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LIBRARY OF CONGRESS. 






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!| UNITED STATES OF AMERICA,|J 



AN 



ELEMENTARY TREATISE 

•fh, 

ON 

ANALYTICAL GEOMETRY 



EMBRACING 



PLANE CO-ORDINATE GEOMETRY 



AND AN 



Jntrokttion to ^ccnutrg of %\xn gimmsians. 



DESIGNED AS A TEXT-BOOK FOR COLLEGES AND 
SCIENTIFIC SCHOOLS. 



BY 

WM. WOOLSEY JOHNSON, B.A., 

Assistant Professor of Mathematics, U. S. Naval Academy. 






PHILADELPHIA 

J. B. LIPPINCOTT & CO 

1869. 



Entered according to Act of Congress, in the year 1869, by 

WM. WOOLSEY JOHNSON, 

In the Clerk's Office of the District Court of the United States, for the District of Maryland. 



Lippincott's Press, 

PHILADELPHIA. 






PREFACE 



Co-ordinate Geometry is the basis of the modern or Analyti- 
cal method in Mathematical Science. Not only does it afford, in 
its applications, the best possible exercises in Algebraic reasoning ; 
but, by familiarizing the mind with the ideas of variables and 
functions, and graphically illustrating them, it is the suitable pre- 
paration for the study of the Differential Calculus and the higher 
branches of Algebra. I have endeavored to adapt this treatise to 
the use, both of those who wish to study the Conic Sections by the 
co-ordinate method, and of those who intend to pursue the higher 
branches of Analysis. 

The common rectangular equations of the Ellipse, the Parabola 
and the Hyperbola have been derived from the usual and familiar 
definitions of these curves ; and demonstrations of their properties, 
founded exclusively upon these equations, occupy the first parts of 
Chapters V., VI. and VII. In this portion of the work, extensive 
application has been made of the ' ; eccentric angle" in the ellipse, 
and, in the case of the hyperbola, of a similar auxiliary angle, of 
which the use is suggested in Salmon's Treatise on Conic Sections. 
In the Chapter on the Hyperbola, the most useful properties and 
equations of the three conic sections have been generalized, with 
especial reference to the manner in which these curves present 
themselves in Astronomy. 

The latter parts of the same Chapters are devoted to more general 
equations of the curves, and to their discussion by means of the 
method of combined equations, which is the basis of the " abridged 
notation." In Chapter VIII., the general equation of the second 

3 



4 PREFACE. 

degree is treated by the method of transformation of co-ordinates. 
It has been my object, in this Chapter, to give methods of deriving 
from an equation in the general form all the circumstances which 
relate to the position and form of the curve ; and also to discuss all 
the special forms of the equation, not examined in the previous 
Chapters. If it be desired to abridge the course by leaving out 
the topics referred to in this paragraph, the student may omit the 
following portions, which are not necessary to the perusal of the 
remainder of the work: Chapter III.; Chapter IV. from Art. 123 
to the end ; Chapter V. from Art 151 ; Chapter VI. from Art. 202 ; 
Chapter VII. from Art. 254; and the whole of Chapter VIII. 

In Chapter IX., I have attempted to classify the principles 
employed in finding the equations of Geometrical Loci, and to 
explain and illustrate them fully by examples solved in the text, 
and a carefully graduated series of examples for practice. Chapter 
X. contains a full explanation of the notation employed in Solid 
Geometry, its application to the plane and the straight line, and 
a brief notice of the various surfaces of the second degree. It is 
believed that this Chapter will be found an adequate preparation, 
in this branch, for the study of standard authors in Analytical 
Mechanics. 

The knowledge of Geometry and Trigonometry essentially pre- 
requisite to the commencement of this study is very slight. I have 
therefore avoided demonstrations founded upon any geometrical 
principles or trigonometric formulae except those fundamental ones 
mentioned in Art. 2. 

Nothing tends to impress mathematical principles and methods 
upon the mind so thoroughly as the solution of numerical examples, 
with their verifications. Such solutions and verifications will be 
found in the text of those Articles in which practical methods are 
explained. Unsolved examples both of a numerical and of an 
algebraic character are added at the ends of the Articles or sec- 
tions. To the more difficult examples, especially to those in which 
algebraic demonstrations are required, are appended hints for their 
solution. 

April, 1869. 



CONTENTS. 



CHAPTER I. 

APPLICATION OF ANALYSIS TO PLANE GEOMETRY. 

PAGE 

Abscissas and Ordinate? 10 

Systems of Co-ordinates 11 

Construction of Points 12 

Negative Values of Co-ordinates 13 

Co-ordinates of Middle Point 14 

Distance between Given Points 15 

Locus of an Equation, or of a Condition 16 

Tracing of Loci by Points 18 

Equation of a Line or Locus of a Moving Point 20 

Algebraic Equations of Lines 21 

Polar Co-ordinates and Equations 22 

Construction of Negative Values of r 24 

CHAPTER II. 

THE STRAIGHT LINE. 

Direction Ratio 27 

Construction of Equations of the First Degree — Intercepts 30 

Intersection of Loci 32 

Combined Equations 33 

Line at Infinity 35 

Arbitrary Constants 36 

Perpendicular Lines 38 

Equation of the Straight Line in terms of its Intercepts 39 

Equation in terms of Perpendicular and Angle 40 

Equations of Condition 43 

Formulae for Straight Lines Passing through Fixed Points 44 

Demonstration of Geometrical Theorems .*. 51 

Polar Equation of Straight Line 53 

Distance of a Given Point from a Given Line 56 

Formula for Line Bisecting the Angles of Given Lines 58 

Equations Representing Two or more Lines 60 

I* 5 



b CONTENTS. 

CHAPTER III. 

TRANSFORMATION OF CO-ORDINATES. 

PAGE 

Formulae for Transformation of Equations 63 

Transformation of a Point — Reverse Formulae 69 

Transformation of Formulae for Lines.. 70 

Arbitrary Transformation 71 

CHAPTER IV. 

THE CIRCLE. 

Circle with Given Centre and Radius 74 

General Rectangular Equation of the Circle , 77 

Circle Passing through Given Points 79 

Polar Equations of the Circle 80 

Intersection of Circle and Straight Line 84 

Condition of Tangency 86 

Tangent to the Circle 87 

Tangents Passing through a Given Point 89 

Polar of a Point with reference to the Circle 90 

General Formula for Polar or Tangent 92 

Length of Tangent from Given Point. 94 

Product of the Segments of a Chord 95 

Intersection of Circles 96 

Radical Axis of Two Circles 97 

Combined Equations of Circles 99 

Properties of a System of Circles with a Common Radical Axis 101 

CHAPTER V. 

THE PARABOLA. 

Definition and Rectangular Equation 103 

Polar Equations of the Parabola 106 

Secant and Tangent Lines — Diameters 108 

Properties of Parabola and Tangent 110 

Oblique Co-ordinates of Parabola Ill 

General Equation when the Axis is Parallel to the Axis of X 113 

Equations of Parallel Lines 114 

Parabola Passing through Given Points 116 

Intersections of Parabolas 118 

Parallel Parabolas 119 

Parabolas Fulfilling Certain Conditions 120 

Equations of the Tangent 123 

Polar of a Given Point 126 



CONTENTS. 7 

CHAPTEK VI. 
THE ELLIPSE. 

PAGE 

Definition and Rectangular Equation 129 

Form of the Ellipse — Relation to the Circle 131 

Polar Equations of the Ellipse 132 

Property of Focus and Directrix 134 

Eccentric Angle 136 

Secant and Tangent Lines 137 

Conjugate Diameters 139 

Equations of the Tangent 140 

Properties of the Ellipse relating to Tangents 141 

Properties of the Ellipse relating to Conjugate Diameters 142 

Lines Bisecting the Angles of Focal Lines 144 

Normal to the Ellipse 146 

Ellipse Referred to Conjugate Diameters 147 

Supplementary Chords 148 

Similar Ellipses 149 

Co-ordinate Axes Parallel to Conjugate Diameters 151 

Similar and Parallel Ellipses — their Intersections, etc 154 

Tangent at a Given Point, and Polar 158 

CHAPTER VII. 

THE HYPERBOLA. 

Definition and Rectangular Equation 162 

Form of the Hyperbola — Asymptotes 164 

Polar and Rectangular Equations involving the Eccentricity 167 

Focus and Directrix of Conic Section — Focal Chords 170 

Conjugate Hyperbolas — Auxiliary Angle 172 

Secant and Tangent Lines 174 

Equations of the Tangent 177 

Properties of the Hyperbola relating to Tangents 179 

Properties of the Hyperbola relating to Conjugate Diameters 180 

Intersection of Tangents with Asymptotes 182 

Tangent and Focal Lines 184 

Equation of the Normal 186 

Hyperbola Referred to Conjugate Diameters 187 

Conic Referred to Axes Parallel to Conjugate Diameters 191 

Product of the Segments of a Chord of any Conic 196 

Intersections of Conies 197 

Property of Chords equally inclined to an Axis 200 

Reciprocal Polars 201 

Hyperbola Referred to its Asymptotes > 203 



8 CONTENTS. 

CHAPTER VIII. 
GENERAL EQUATION OF THE SECOND DEGREE. 

^ PAGE 

Criterion Distinguishing the Three Conies 208 

Change of Origin 209 

Condition for which the Conic becomes a Pair of Straight Lines 211 

Change in Direction of Axes 212 

The Central Equation 216 

The Conic Referred to a Tangent 220 

Tangents and Diameters 223 

Rectangular Equations 226 

Conditions of the Circle 227 

Direction of the Axes of Conic from Rectangular Equation 228 

Direction of the Axes of Conic from General Equation 229 

Semi-axes of Conic from Rectangular Equation 230 

Semi-axes of Conic from General Equation 231 

Conic Fulfilling Given Conditions 232 

General Equation of a Polar 238 

CHAPTER IX. 

GEOMETRICAL LOCI. 

Choice of Co-ordinate Axes 240 

Application of Analytical Formulae..... 243 

Elimination of Variables 245 

Intersection of Variable Lines 348 

Locus of a Point Connected with a Variable Line 250 

Use of Polar Co-ordinates 252 

CHAPTER X. 
APPLICATION OF ANALYSIS TO SOLID GEOMETRY. 

Co-ordinate Axes and Planes 255 

Co-ordinates of Direction — Spherical Co-ordinates 257 

Polar and Rectangular Co-ordinates 258 

Method of Projections 260 

Direction Angles 262 

Angle between Given Directions 265 

Transformation of Co-ordinates 267 

Equations between Co-ordinates — Surfaces and their Sections 271 

Equations of the Plane 273 

Equations of the Straight Line. 278 

Distance of a Point from a Given Point, Line or Plane 283 

Surfaces of Revolution 284 

Ellipsoid, etc 285 



ANALYTICAL GEOMETRY. 



CHAPTER I. 

APPLICATION OF ANALYSIS TO PLANE GEOMETRY. 

Art. 1. Quantity is that which can be measured by a unit of 
its own kind ; as length, time, weight. A magnitude of any kind 
is represented by the number of units it contains. Algebra, the 
science of quantity in general, treats of number as the representa- 
tive of magnitude, and its processes are applicable to all kinds of 
quantity. 

Analysis is the term applied in mathematics to that treatment 
of a subject, in which appropriate magnitudes, represented by sym- 
bols, are introduced into algebraic equations and subjected to alge- 
braic processes. 

It is the object of this treatise to explain a system by which 
Plane Geometry is subjected to this treatment. 

2. This system is based upon a few fundamental principles 
of Greometry ; namely, the doctrines of parallel lines and of similar 
triangles, and the property of a right-angled triangle, that the 
square of the hypothenuse is equal to the sum of the squares of 
the sides. 

The method of investigation presupposes a knowledge of 
the principles of Algebra, so far as necessary to the solution 
and discussion of equations of the first and second degrees; 
of the usual notation of Trigonometry for angles and their 
functions, sin, cos, etc.; and of the fundamental relations, 



10 APPLICATION OF ANALYSIS. 



— = tan, sin 2 -f- cos 2 = 1, etc., which are consequences of the 

definitions of the functions and of the property of a right triangle 
above referred to. 



Position. 

3. The magnitudes which, in the system to he explained, enter 
into algebraic equations are those which determine the position of 
a point in a plane. 

Let OX, OY be two fixed intersecting lines of the plane, and 
P be any point. From P let PR and 
PS be drawn, each parallel to one of 

the fixed lines and terminated by the fe /p 

other; then the distances PR and PS 
expressed in terms of some unit of 
length determine the position of P. 

The fixed lines of reference, OX and 
OY, are called the axes, and the lengths 

PS and PR are called the co-ordinates of the point P. The 
former is distinguished as the abscissa and the latter as the ordi- 
nate, and the axes to which they are parallel are called respectively 
the axis of abscissas and the axis of ordinates. It is custom- 
ary in the figures to draw the former horizontal or parallel to the 
bottom of the page, so that OX is the axis of abscissas and OY, 
of ordinates. 

A point is said to be given, when the number of units or parts 
of a unit contained in its abscissa and ordinate are given; these 
numbers are themselves called the co-ordinates of the point. Thus, 
a point is given, when we say its abscissa is 3 J, and its ordinate 
is 2. 

4. Since SPRO is a parallelogram, OR = SP, and therefore 
OR may be taken as the abscissa of P. For the same reason, 
it is the abscissa of every point on the line PR, produced in- 
definitely either way ; that is to say, every point of a straight line 
parallel to OY, the axis of ordinates, has the same abscissa. In 
like manner, every point of the line SP parallel to the axis of 
abscissas has the same ordinate, and OS might be regarded as 



SYSTEMS OF CO-ORDINATES. 11 

their common ordinate ; but it is usual, of the distances PR and 
PS, to draw only the former to represent the ordinate, letting OR, 
the part of the axis of abscissas cut off, represent the abscissa. The 
ordinate of every point in the line OX is evidently zero, and its 
abscissa is its distance from the point 0. The ordinate of every 
point in the line OY is also its distance from 0, while its ab- 
scissa is zero. The point where the axes intersect is called the 
origin; each of its co-ordinates is zero. 

5. The symbols used to denote the co-ordinates of a point are 
those usually devoted in algebra to unknown quantities — namely, x 
and y, because, in the class of problems we shall first meet, they 
will be the unknown quantities. By common consent, x has been 
appropriated to the abscissa, and y to the ordinate. For this 
reason the axis of abscissas is marked in the figures by a capital 
X, and is frequently called the axis of X ; and the axis of ordi- 
nates is distinguished by a capital Y, and called the axis o/Y. 
The abscissas of known points will usually be denoted by such 
symbols as as', x" , x 1 , x 2 , etc., the ordinates by y', y" , y x , y 2y etc., and 
the points to which these co-ordinates refer, correspondingly, by 
F, P", P x , P 2 , etc. 



Systems of Co-ordinates. 

6. It is evident that one of the co-ordinates alone will not define 
completely the position of a point; and indeed, owing to the dimen- 
sions of space, it is impossible that one numerical element should 
determine position in a plane. To define position upon a line, we 
would need to assume a fixed point of reference on the line, or 
origin of distances, as well as a unit of distance, and then the posi- 
tion of a point would be defined by the number of units in its dis- 
tance from the origin ; just as to define an instant of time, we as- 
sume a fixed instant or epoch of reference, and a unit of duration. 
But, if the point is not confined to a certain line, but only to a sur- 
face in which it is free to move in all directions, two determining 
elements are required; and if it is not limited at all, but free to 
take any position in space, three are necessary. These determining 
elements, of whatever kind they may be, have received the general 
name of co-ordinates. In treating of the relative positions of bodies 



12 APPLICATION OF ANALYSIS. 

in space, or questions of " Solid Geometry," the three dimensions 
of space render it necessary to use three co-ordinates, but in Plane 
Geometry we need two, and two only, because all the points treated 
are confined to a single surface. As an example of the same prin- 
ciple, we may observe that the position of a point on the earth's 
surface is defined by two co-ordinates, its latitude and longitude ; 
while, to define completely the position of any point relatively to 
the earth, we need a third co-ordinate, as height above the level of 
the sea, or distance from the earth's centre. 

7. Different systems of co-ordinates for position in a plane may 
be proposed. The system we have just described, in which the 
point is referred to two fixed lines, is the principal one in use. It 
is called the system of Cartesian co-ordinates, being the invention 
of Des Cartes. Besides this, there is another method in use called 
" Polar co-ordinates," in which the position of a point is defined by 
its distance from a fixed point called the pole, and the direction in 
which this distance is measured — the co-ordinate determining the 
latter being an angle. Cartesian co-ordinates are called rectangu- 
lar or oblique, according as the axes are at right angles or obliquely 
inclined. Rectangular co-ordinates is a special case of oblique, in 
which the axes may be inclined at any angle whatever. Formulae 
are frequently very much simplified by their use, but care must be 
taken not to extend to the general case those which apply only to 
rectangular co-ordinates. 

Construction of Points. 

8. Suppose now, we are required to find the point whose abscissa 
is 7, and whose ordinate is 5. The 
axes being drawn and a unit of 
length assumed, we might lay off the p' 
abscissa on the axis of X, either on 
the right or on the left of the origin ; 
it is customary to lay it off to the 
right, and to lay off the ordinate up- 
ward. Therefore, laying off seven 
of the assumed units on the axis of 

X, from the origin toward the right hand, and from the point so 
reached five units upward on a parallel to the axis of Y, we find 




CONSTRUCTION OF POINTS. 13 

the required point. We should have arrived at the same point 
had we first laid off five units on the axis of Y, and then seven 
units on a parallel to the axis of X. The operation is called the 
construction of a given point, and the point is referred to as the 
point seven, five, and written thus (7, 5), the values being enclosed 
in brackets, and that of x written first. The construction of a 
point consists of two parts ; in the process used above, we first con- 
structed the value of the abscissa or x, by constructing the point 
(7, 0) and drawing the parallel, for every point of which x = 7; 
we then construct the value of y, by finding the point of this line, 
for which y = 5. Or we might construct the line for every point 
of which y = 5, and then find the point of this line for which 
x = 7 ; that is, the point common to the two lines. 

9. From the rule to lay off the value of the abscissa to the right, 
it evidently follows that adding to the value of x carries the point 
to the right, and subtracting from it, to the left. Adding a number 
of units to the value of x (leaving that of y unchanged) is there- 
fore equivalent to moving the point P so many units to the right 
along the dotted line PP\ We can thus increase the value of x, 
or conceive it increased to any extent. Subtracting from the value 
of x, would move P on the same line an equivalent number of units 
to the left, which motion would decrease the value of x, until, hav- 
ing subtracted the whole value of x, we arrive at the point S, whose 
abscissa is 0. But as motion to the left can take place indefinitely, 
as well as motion to the right, we can, in this sense, subtract from 
the value of x a greater number of units than it contains. Thus, 
moving the point P, of the figure, 10 units to the left, we arrive at 
the point P', whose abscissa is 7 — 10. The subtraction indicated is 
arithmetically impossible, but in algebra the result is — 3, which 
therefore properly expresses the value of the abscissa of P', a point 
3 units to the left of the axis of Y. Addition and subtraction, be- 
ing thus represented by two opposite and mutually destructive 
motions, are always equally possible ; subtraction no longer implies 
the existence of some quantity to subtract from, and the " negative 
quantities" of algebra become equally intelligible with the "posi- 
tive." 

We may consider motion to the left as still decreasing the value 
of x, even when the point is already on the left of the axis of Y. 



14 APPLICATION OF ANALYSIS. 

This sense of the word decrease will be distinguished from the 
ordinary one as algebraic decrease ; so that, for a point on the left 
of the axis, motion to the left ' ; algebraically decreases," though 
numerically increasing, the negative value of x. 

In the same mauner it may be seen that increase and decrease in 
the value of y correspond to motion upward and downward, and 
that a negative value of y properly expresses position below the 
axis of X. Thus we can construct a point with any given values 
whatever, positive or negative, laying off the positive values to the 
right and upward, the negative to the left and downward. It is, 
of course, arbitrary which directions we regard as positive, but 
whichever we select, the opposite must be considered negative. 
The capitals X and Y, in the figures, will always be placed on their 
respective axes in the positive directions from the origin. 

Relative Position of Points. 

10. As the position of a point depends upon the values of its 
co-ordinates, so the relative position of two points depends upon 
the differences of their co-ordinates. If, in passing from one posi- 
tion to another, the co-ordinates are both increased, the point moves 
upward and to the right ; that is, it follows a direction intermediate 
to the positive directions of the axes. If they are both diminished, 
the direction is the reverse of this ; if x is diminished and y in- 
creased, it passes upward to the left, etc. In general, let P' and P" 
be two given points, of which we 
wish to know the position of P" 
relatively to P'. We have only to 
notice the signs of the differences 
x" — x' and y" — y', which are 
the values of P'N and NP". In 
the figure they are both positive. 
Comparing P' with P", the differ- 
ences x r — x" and y' — y" are 
both negative. These differences 
evidently determine both the rela- 
tive direction of P' and P" and the distance between them. 

11. To find the co-ordinates of a point midway between two 
given points. Let x' y f , x" y", be the co-ordinates of the given 



Y 




¥/ 


P" 
i 




v'Z. 


M" 


...Jn 
j 









■ 











RELATIVE POSITION OF POINTS. 15 

points, and let x and y be the co-ordinates of the point P. bisect- 
ing the line P'P". Then P'M : FN : : P'P : P'P" : : 1 : 2, by simi- 
lar triangles ; that is, P'M = \ P'N. But x — x' represents 
the length of P'M, and x" — x' , of P'N, therefore x — x' = 
i (x" — cc'). Similarly we prove y — y'=i(y" — y') • hence 

x = i (x' + x") and y = § (jf -f y"). 

These results may be expressed thus : Each of the co-ordinates of 
the middle point is an arithmetical mean between the correspond- 
ing given co-ordinates. In deriving results, as above, from a figure 
and a geometrical property, the algebraic expressions put in the 
place of lines should all be positive. 

12. The distance of given points, as P' and P", depends upon 
the lengths of P'N and XP", and also upon the included angle 
P'NP". If the axes are rectangular, the square of the distance 
P"P' equals the sum of the squares of these lines, hence 

P' P" = yV'-*') 2 + (/'-/) 2 

is the formula for the distance of two points, when the axes are 
rectangular. As this formula is simpler than the corresponding 
one for oblique co-ordinates, which involves the angle between the 
axes,* questions in which the distance between points is concerned 
are usually treated by rectangular co-ordinates. 

Examples. — Construct the points ( — 1.3) and (3. — 5), 

Determine the co-ordinates of the point bisecting their distance. 

Find the distance of each from the middle point, the axes being 
rectangular. 

What are the co-ordinates of the point bisecting the distance of 
P' from the origin, and the formula for that distance ? 

Ans. x = ^y = l l OP' = l/V 2 + j/\ 

* If the axes be oblique, P'XP" is the supplement of YOX. If then 
YOX (the angle between the positive directions of the axes) be repre- 
sented by cj, trigonometry gives 

P'P" = V ( x " — x' ) 2 4- (y" — y') 2 -f 2 (*" — x') {y" — y') cos oi, 
which reduces to the above form, when o> = 90°. 



16 APPLICATION OF ANALYSIS. 



Locus of an Equation. 

13. We have seen that two co-ordinates or determining elements 
must be known in order to fix the position of a point. These may 
he given directly, in algebraic language, by two equations, such as 
x = 7, y = 5 ; or indirectly by two equations between x and y : as 
x-\-y = \2,x — y = 2. from which the above values may be de- 
duced. The equations x = 7, y = 5, taken together, are called 
the equations of the point. It was shown, in Art. 8, in which this 
point was constructed, that the equation x = 7, taken alone, enables 
us to construct a line parallel to the axis of Y, on which the point 
is to be found, because it contains all the points for which x = 7. 
This equation, therefore, or the single 

condition that its abscissa be 7, limits V \ 

the point to a particular line, though \ \ 

leaving it yet undetermined. The sin- 

gle condition y = 5 also limits the 
point to a certain line, namely, a paral- 
lel to the axis of X; but the two con- — 
ditions, taken together, determine the 
point as the one common to the two 
lines, because that is the only one 
which fulfils at the same time both conditions. 

14. It will be seen hereafter, that each of the equations 
x -J- y — 12 and x — y = 2, is a condition which taken singly 
limits the point to a certain line, and that P, the point (7, 5), is the 
common point or intersection of these two lines. In general, every 
equation between x and y is a condition imposed upon the point to 
which it refers, which limits it to a certain line ; this line is called the 
locus of the equation, that is, the place to which it restricts a point. 

For example, take the simple equation x = y. This expresses 
the condition that the abscissa and ordinate of the point are equal. 
It evidently does not determine the point, because it is true of a 
variety of points as (1, 1), (2, 2), etc. Let a straight line be 
drawn through the origin bisecting the angle YOX ; it will readily 
be seen that the abscissa and ordinate of every point of this line 
are equal, and that this is true of no other points; therefore the line 
is the locus of the equation x = y. 




INDETERMINATE EQUATIONS AND FUNCTIONS. 17 

In like manner construct the locus of the equation x = —y, which 
expresses the condition that the abscissa and ordinate be equal with 
contrary signs. 

15. Whenever, in Plane Geometry, a point is to be determined, 
there must be given two independent conditions. Each condition 
taken by itself limits the point to a certain line, which may be 
called its locus. When the loci of the two conditions are drawn, 
the solution is completed, for the common point or points of the 
two loci can alone fulfil both the conditions. Thus, if we wish to 
describe a circle through three given points of a plane, A, B and 
C, we may find its centre by the independent conditions, that it 
must be equally distant from A and B, and also that it be equally 
distant from A and C. A perpendicular 
bisecting AB is the locus of the first 
condition, because it can be proved that 
it contains all the points which fulfil that 
condition, and no others. A perpendicu- 
lar bisecting AC is the locus of the other 
condition, and as the loci have but one 
common point P, there is one and but 
one solution. If A,B and C are in one 
straight line, the loci will be parallel, and there will be no solution. 

The above may be called a graphic solution, because the loci of 
the conditions are drawn. In the analytic method, the conditions 
of a problem are thrown into the form of equations between the 
co-ordinates of the point sought, whose values are then found by 
the algebraic process of elimination. 

Indeterminate Equations and Functions. 

16. Since two conditions are thus necessary to determine a point 
in a plane, if but one is given, the problem is said to be indeter- 
minate. So in algebra a single equation between two unknown 
quantities is called " an indeterminate equation/' because insuffi- 
cient to determine their values. Such an equation between x and 
y may in general be satisfied by a value of x, selected at pleasure, 
and a-corresponding value of y determined from the equation; thus 
x = 2y — 3 is satisfied by x = and y — 1-J-, x = 1 and y = 2, 
x = 2 andy — 2-J-, etc. Regarding, then, x as independent and 




18 



APPLICATION OF ANALYSIS. 



y as depending upon it, we see that the equation may be satisfied 
by an indefinite number of sets of values of x and y. A quantity 
depending upon another for its value, as y does here upon x, is 
said to be a function of it. The unknown quantities, x and y, of 
an indeterminate equation are called variables ; because one of them, 
as x, may be regarded as capable of assuming any number of dif- 
ferent values or varying independently, while the other, in order to 
satisfy the equation, must take certain corresponding, values, and 
therefore varies dependency upon the former. 

A function is said to be explicit when its value is directly ex- 
pressed in terms of the independent variable, but implicit whsn 
the dependence is merely implied by the existence of an equation. 
For instance, if x 2 -\- y 2 — 25, y may either be regarded as an 
implicit function of x, or as the independent variable of which x 
is an implicit function. Solving for y, we have y = ±- l/25 — x 2 , 
in which y is an explicit function of x. 

Examples. — Find sets of values satisfying x -j- y = 12; sets of 
values satisfying x — y = 2. 

Make y an explicit function in each case. 

17. Let x and y, in an indeterminate equation, stand for Cartesian 
co-ordinates ; then, for each of the sets of values of x and y, a point 
may be constructed. Each of these points will be a point of the locus 
of the equation, and is said to satisfy the equation, because its co- 
ordinates satisfy the equation. The equation of the last article, 
x = 2y — 3 is satisfied by the points (0, 11), (1, 2), (2, 21), etc. 
In order conveniently to find a 
number of points of the locus 
of an equation, make y an ex- 
plicit function of x ; thus y = 
l (x -\- 3) ) then giving to x a 
number of equidistant consecu- 
tive values, as 0, 1, 2, 3, 4, 5, 
the corresponding values of y, 
1-J-, 2, 21 3, 31 4 are readily 
found, and points P, P, P con- 
structed as in Art. 8. It is evident. in this case, from the value of y 
in terms of x, that any increase in the value of x produces an increase 
of half the amount in y. As a consequence of this, the points con- 




INDETERMINATE EQUATIONS AND FUNCTIONS. 19 



-.JP 



structed lie on one straight line, as in the figure. This straight 
line is in fact the locus of the equation, but neither these points, 
nor any number of others which might be constructed with inter- 
mediate values of x, properly speaking, constitute the line. It 
may, however, be conceived as described by a moving point whose 
abscissa increases uniformly from the value zero, while its ordinate 
increases from the value 1-J-, also uniformly, but at half the rate. 
If x pass through all possible values, positive and negative, the 
whole line will be described. 

18. The law of the variation of a function is not generally so sim- 
ple as in the above case. Take, for instance, the equation x 2 -j- jf 
= 25. Making y an explicit function, we have y = ± j/25 — x 2 , 
in which if we give x the values 0, 1, 2, 3, 4, 5, we obtain for y 
.the series of values, ± 5, ± l/2i (about 4.9), ± i/21 (about 4.6), 
±4, ±3, 0. The positive values of y in this case decrease as the 
values of x increase, and the rate of decrease is not uniform, but 
becomes more and more rapid. If the abscissa of a point increase uni- 
formly from zero, while its ordinate de- 
creases from 5, being always the above 
function of its abscissa, it will describe 
a curve, because the rate of y is not 
uniform. If this curve were drawn, it 
would therefore represent the function, 
and its direction at any point would 
show the rate of decrease in y. For 
this reason, a number of points are fre- 
quently constructed for a function, and 
a line roughly traced through them to 
represent the function approximately; 
usually taken rectangular. 

19. Constructing in this way the values of y, obtained above, for 
the function y = l/25 — x 2 , and tracing a line, we should hardly 
fail to notice that it approximates, to the quadrant of a circle. Be- 
cause the value of y decreases as that of x increases, it is called a 
decreasing function ; the rate of decrease is variable, being evidently 
greater for a greater value of x. In the case of Art. 17, as y in- 
creased with x, the function was an increasing one. If the circle, 
with centre at the origin, and radius five units in length, be drawn, 



•-•JP 



the axes in such a case are 



20 APPLICATION OF ANALYSIS. 

we can prove that the co-ordinates of each of its points satisfies the 
equation x 2 -J- y 2 = 25 ; because they form the sides of a right tri- 
angle whose hypothenuse is the radius. All the quadrants of the 
circle are included, because the positive values of x give also nega- 
tive values of y, and the corresponding negative values of x give 
the same positive and negative values of y. This circle, therefore, 
in the sense explained, represents the equation re 2 -j- y 2 = 25, 
which is therefore said to be the equation of this circle. 

Examples. — Trace the lines representing the functions, y ==. 
x — 2, y = x 2 , y = 5 — 2x. 

Trace the locus of x 2 — y 2 = 16 ; of x 2 -\- y 2 = 36. 

Equation of a Line or Locus. 

20. If a point move under a certain condition, or according to a 
certain law, it will describe a line. The equation, of which this 
line is the locus, expresses the condition or law, in the form of a 
relation between the co-ordinates of the point. The line will be 
called the locus of the point moving under the law, as well as the 
locus of the condition or of the equation which expresses it, and 
the equation will be called the equation of the line. Thus, x = y 
is the equation of a straight line bisecting the angle between the 
axes, or of the locus of a point, so moving as to be always equidistant 
from the two axes ; x 2 -j- y 2 = 25 is the equation of the locus of a 
point always 5 units distant from the origin. The equation of a 
line need not be written so as to make y an explicit function of x, 
for it connects x and y together, so that either may be considered 
a function of the other. If we wish to ascertain whether a given 
point is on the line of which we have the equation, we have merely 
to substitute the given values for x and y, and see if they satisfy 
the equation.. Thus (7, 5) is on the line represented by x -\- y 
= 12, but (8, 3) is not. 

21. As every regular or mathematical line or curve may be de- 
scribed by a point moving according to some law, every such line 
must have an equation. The law of the motion is equivalent to 
some common property of every point of the line, which may be 
made the definition of the line; as in the case of the circle above, 
the common property of whose points is their distance of 5 units 
from the origin. The equation x 2 -j- y 2 = 25 expresses a common 



ALGEBRAIC EQUATIONS OF LINES. 



21 



property of all the points, which is proved to be equivalent to this 
defining property. We have now to show how any law of motion, 
or common property of points, may be converted into an equivalent 
relation between co-ordinates; that is, into an equation between 
x and y. 

22. Given the law of motion of a describing point, or a common 
property of all the points of a line, such as is involved in its defi- 
nition, we proceed as follows : Construct a figure, (in which let P 
represent any point of the line, or the moving point in any of its 
positions,) by drawing the lines referred to in the conditions of the 
problem. Select lines as the axes of co-ordinates, and draw the 
co-ordinates of P, which mark by the letters x and y. These are 
called the general co-ordinates of the line, because P does not re- 
present a particular point, but any point of the line. Then, by 
means of geometrical principles, establish a relation between x and 
y. As this relation must be true of every point of the line, it is 
the equation required. 

For example, suppose P to move so that the sum of the squares 
of its distances, PA and PB, from two fixed points, A and B, shall 
be constantly 82, the distance be- v 

tween A and B being 8. Let 
the line AB be selected as the 
axis of X, and let the middle 
point of AB be the origin, and _ 
the axes rectangular. By the 
condition PA 2 -f PB 2 = 82, and 
by right-angled triangles, since 
'AO and OB each equal 4, PA 2 = 

f -j- (4 + x)\ PB 2 = if -f (4 — x) 2 ; hence adding, 2/ + 
(4 _i_ x y + (4 — x) 2 = 82. Reducing, x 2 -L- y 2 = 25 is the 
equation required. 

Example. — Find the equation, when the condition is that 
PA 2 — PB 2 = 48. 



Algebraic Equations of Lines. 

23. It is not necessary that the line of which the equation is 
sought should itself be drawn in the figure; on the contrary, its 
form may be entirely unknown, until revealed by the equation found, 



22 APPLICATION OF ANALYSIS. 

as in the illustration given in the last article. For, x 2 -j- y 2 = 25 
having been already found as the equation of a certain circle, we 
now know the form of the line described by P under the proposed 
condition. 

When the form of the line is known, we take P anywhere on the 
line at random, and make use of the denning property of the line 
to establish the relation between x and y. 

24. We shall, with the next Chapter, begin systematically to 
develop the equations of known lines in various positions, and to 
classify them by their equations, carrying this so far as to be able, 
on resuming, in Chapter IX., the consideration of " Geometrical 
Loci,." to ascertain the position and form of the line represented by 
any equation of the first or second degree. 

W T hen the constant quantities which enter into the equation of 
a line are represented by letters to which any values may be 
assigned, the result is an algebraic or general equation, which in- 
cludes a variety of numerical equations, and hence may represent 
any one of a certain class of lines. Thus, x = 7 was found to re- 
present a parallel to the axis of Y, at a distance of seven units on 
its right; but x = a may be made, by giving different values to a 
to represent any parallel to that axis ; it is therefore said to repre- 
sent generally a parallel to the axis of Y. Similarly, y = b repre- 
sents a parallel to the axis of X. Suppose now we had found an 
algebraic equation or formula, which, by giving different values to 
the constants, might be made to represent a straight line in any 
position whatever. Then every equation which can be reduced to 
this form will be known to belong to some straight line, and the 
general form, of the equation so reducible will be the general equa- 
tion of the straight line. The algebraic equations also enable us to 
develop the properties of lines in a more general manner. 

Polar Co-ordinates. 

25. We shall now describe the other system alluded to in Art. 7, 
in which distance from a fixed point, and direction, are used to de- 
termine position. 

Let be a fixed point, and OA a straight line drawn from it 
toward the right, and let P be any point of the plane. is called 




POLAR CO-ORDINATES. 23 

the pole, and OA is called the initial line, as it marks an initial 
direction, with which all other directions are compared. 

Join OP, the position of P is evi- 
dently determined by the length of OP 
its distance from the pole, and the 
value of the angle POA. The dis- 
tance OP is called the radius vector, 
and denoted by the symbol r, and the 

angle POA is denoted by the Greek letter ; r and taken 
together are the Polar co-ordinates of P. The angular co-ordinate 
is reckoned from OA, over toward the left, as in the figures in 
trigonometry. The value of in the figure is less than 90° ; the 
value = 0° would indicate that the radius vector had the 
direction OA, or that P was on the line OA, and if be increased 
from that value, r remaining constantly the same. P will describe a 
circle with centre at the pole, completing it and returning to its 
primitive position, when = 360°. 

The direction of the point P being thus completely represented 
by the value of 0. its distance, r may be regarded as essentially posi- 
tive. If r = 0, P is at the pole, whatever the value of 0, and by 
increasing r without limit, we may define any point of the plane by 
a positive value of r and a value of between 0° and 360°. The 
method of constructing a point with given co-ordinates is simply to 
draw a straight line from O with the direction marked by 0, and 
then measure off on it from O a number of units equal to the value 
of r. 

26. The values of two co-ordinates being necessary in this sys- 
tem to determine the position of a point, (except of the pole itself, 
which is determined by r = 0,) one given co-ordinate, or one equa- 
tion between co-ordinates, restricts the point to a certain line or 
locus, as in the Cartesian system. Thus, r = 5 taken alone re- 
stricts the point to a circle with radius 5, and its centre at the pole ; 
.0 = 40° restricts the point to a straight line, having this inclina- 
tion to the initial line, and terminated in one direction by the 
pole. 

In an equation between r and 0, r should be regarded as a func- 
tion of 0. Then by assuming values of 0, and determining from the 
equation the corresponding values of r, any number of points may 



24 APPLICATION OF ANALYSIS. 

be constructed which satisfy the equation. An approximation to 
the locus of the equation, may then be constructed by tracing a line 
through the constructed points ; but it may be conceived as de- 
scribed by continuous motion, if a point P on a line uniformly re- 
volving about the pole move in that line according to the law of 
the function ; that is, so that its distance r from the pole shall 
always be the proper function of the angle 0, described by the line. 

27. Equations between r and 6 are of two kinds — those in which 
the angle 6 occurs only through its trigonometric functions, as 
r cos 6 = 7, and those in which the angle itself occurs, as r = 30. 
In the first, r depends only upon the direction marked by 6. The 
value of 6 may then be expressed in degrees, and no distinction is 
to be made between values which differ by 360°, as 10° and 370°, 
because they mark the same direction, and have the same functions. 
But in the other case is expressed in the arcual or circular mea- 
sure, in which 2n, the length of the circumference of a circle whose 
radius is the unit, takes the place of 360° ; and values which differ 
by 2?r, though marking the same direction, will not give the same 
value of r. Thus, in r — 30, the value of r uniformly increases 
from to 67r, during the first revolution of the line ; that is, while 
6 increases from to 2n ; and then, during the second revolution, 
continues to increase from 67r to 12tv, while must be considered 
as passing from 2tz to 47r. Thus the value of r may be increased 
indefinitely; the curve described is called the "Spiral of Archi- 
medes." 

Polar equations of the first class alone will be treated. 

28. It may happen in a polar equation that some values of 
have, corresponding to them, negative values of r. Although nega- 
tive values of r are not required, to 

define the position of certain points, 
as negative values of x and y were 
in the other system j yet such cor- 
responding values of 6 and r may 
be constructed. Thus, suppose \ r = — 5 

6 = 100° gives for r the value \ 

— 5, we draw a line from O, making p 

this angle with OA, and then measure off 5 units on the line pro- 
duced through O. This is consistent with the ordinary interpreta- 



POLAR CO-ORDINATES. 25 

tion of negative values, and the consideration of such values may be 
necessary to the complete discussion of an equation. 

Evidently the same point would be denoted by an equal positive 
value of r, and 6 = 280°. By admitting negative values of r, we 
might dispense with all values of 6 above 180°; then the whole 
line, drawn in the figure, would be represented by 6 = 100°, the 
part below the initial line containing points for which r must be 
taken as negative. 

We may adopt a notation for a point given by polar co-ordinates 
similar to that of Art. 8, (7, 100°), denoting the point whose radius 
vector is 7 units and its inclination 100° ; then ( — 7, 280°) denotes 
the same point. 



CHAPTER II. 



THE STRAIGHT LINE. 



29. In order to find the algebraic equation of the straight line, 
we must draw a representative line, unrestricted in its position with 
respect to the axes, and establish an equation between the co-ordi- 
nates of a point, taken at random on the line, and certain con- 
stants depending upon the position of the line. The equation must 
be true for every point of the line drawn, with the same values of 
the constants; that is, true for the different values of x and y cor- 
responding to all points of the line, otherwise it does not represent 
the line. But different values being given to the constants, the 
equation must be capable of representing any straight line, other- 
wise it will not be what we are seeking — namely, a general formula 
for the straight line. 

30. But, first let us con- 
sider a particular class of 
straight lines — namely, those 
which pass through the ori- 
gin. Draw such a line, and 
the ordinates of several of 
its points. The triangles 
PRO thus formed are all 
similar, and therefore the 
ratio PR : RO or y : x is 
constant for all points of the 
line. Let this ratio or the 

y 
quotient - be represented by ra; then we have for the equation 

V 

- = m,or y = mx. 




The value of the constant ratio, 

26 



determines the direction of the 



THE STRAIGHT LINE. 27 

line, it is therefore called the direction ratio. In the figure, m is 
positive, for the co-ordinates x and y are either both positive or 
both negative, so that their quotient is always positive. 

If the line had been drawn in a direction intermediate to the 
positive direction of one axis and the negative direction of the 
other, its points would have co-ordinates of different signs, and the 
value of m would be negative. 

Whatever the value of m, the origin satisfies the equation 
y = mx ) that is, the origin is always a point of the line. To 
construct the line with a given value of m, it is only necessary to 
construct one other point satisfying the equation ; thus, if m = 2, 
that is, if we have the equation y = 2x, assume any value of x, as 
35 .= 1, then y = 2, and the point (1, 2) is a point of the line. 
Joining this point with the origin we construct the line. 

Hereafter we shall refer to a line given by its equation, as the 
line y = 2x, the line x rf- % = 3, etc. 

Examples. — Construct y = 5x, x -f- y = 0, x = 2y. 

Give the value of m in each case. 

31. Evidently the smaller the value of m the more nearly does 
the direction of the line y = mx approach to that of the axis of X. 
If we give m the value zero, the equation becomes y = 0, which 
represents that axis itself, for on the axis of X the ordinate is 
always zero. 

1 x 

Let n = — : then n is the value of the ratio — , and x = ny 

111 . y . 

represents the same line as y = mx. Now, suppose a line pass- 
ing through the origin to revolve from right to left, and con- 
sider its equation both in the form y = mx and x = ny. When 
the line coincides with the axis of X, m = 0. During the revo- 
lution the value of m increases and that of n decreases. As 
the line approaches the axis of Y, m becomes very great and n very 
small, and finally when it coincides with the axis of Y, n = 0, and 
m is said to be infinitely great or simply infinite. If the revolution 
be continued in the same direction from this position, n becomes 
negative and numerically increases, so that it still decreases alge- 
braically; m also becomes negative and numerically decreases, so 
that it continues to increase algebraically until the line again coin- 
cides with the axis of X. when m = and n is infinite. In gene- 
ral m and n have definite values, which are reciprocals of each 




28 THE STRAIGHT LINE. 

other, and either form y = mx or x = ny may be used ; but in 
the special cases in which one of these quantities is zero, the other 
is infinite, and but one form can be used, the resulting equations 
being y = and x = 0,the equations of the axes. 

32. Now let the line 
BP be drawn cutting 
the axis of Y in any 
point, B. Draw OP 
through the origin par- 
allel to it, and let m 
represent the direction 
ratio of OP'. Draw- 
ing the ordinate of any 
point, P, we see that it 
consists of two parts, 

P'R the ordinate, corresponding to OB, in the line OP', and PP', 
which, by the properties of the parallelogram, is equal in all cases 
to OB. Representing this part of the ordinate, which is constant, 
by b, we have 

y — mx -f- b, 

for the equation of the line. As BP has the same direction as 
OP', m is its " direction ratio," and any change in the value of m 
would have the same effect on the direction of the liney = mx -f- b, 
as on y = mx. When m is positive, as in the figure, mx will be 
negative for negative values of x, so that y is less than b for points 
on the left of the axis of Y. For a certain negative value of 
cc, mx will be numerically equal to b : and y will be zero, and for 
greater negative values of x, y will be negative. But when the 
value of m is negative, mx is negative for positive values of #, and 
y is less than b for points on the right of the axis. The direction 
of the line being determined by m, the value of b determines the 
position of the line, by determining the position of one of its points, 
B, and it is plain that by giving different values to 6, we may pro- 
duce the equation of any line parallel to y = mx. When b = we 
have the line y = mx itself, and when b is negative the line lies 
below OP', and the part PP' is subtracted from P'B. Thus, 



THE STRAIGHT LINE. 29 

y = 2x -\- 2, y = 2x -j- 1, y = 2x, y = 2x — 1, etc., is a series 
of parallel lines whose direction ratio is 2. 

33. In y = mx -j- &, ^ is an explicit function of x, and the 
equation may be regarded as expressing that when x = 0, y = b, 
and that as x increases uniformly from the value zero, y increases 
uniformly from the value 6, but at the comparative rate m. This 
is but a general statement of what was said in Article 17 of the 
equation x = 2y — 3, which was put in the form y = ix -\- 1^-, 
in which — | — Jr is the value of m, and expresses the comparative 
rate of the variation of y, and 1-1- is the value of y corresponding 
to x = 0. 

Every equation of the first degree will take the form y = mx -f- b, 
if we make y an explicit function of x ; and since every equation of 
this form represents some straight line, an equation between x and 
y of the first degree must represent a straight line. For this rea- 
son such equations have been called linear equations, and functions 
of the form y = mx -f- 6, linear functions. 

The equation of a straight line may also be written, 

x .== ny + a, 

in which x is made an explicit function of y. This is the equa- 
tion of a line parallel to the line x = ny (which passes through 
the origin), and cutting the axis of X at the distance a from the 
origin to the right or left, according as a is positive or negative. 

34. If the line is oblique to both the axes, we may express its 
equation in either of the forms y = mx -\- b, or x = ny -j- a, but 
if it is parallel to the axis of X, m = 0, and the first form gives 
for its equation 

y = b. 

If it is parallel to the axis of Y, n = 0, and the second form re- 
duces to 



In these cases the co-ordinates are not variables, either of which 
is a function of the other, but one of them is constant in value, in- 
dependently of the other. 

35. The general equation of the first degree, 



3 * 



30 THE STRAIGHT LINE. 

Ax -f By + C = 0, 

consisting of a term containing x, a term containing y, and a term 
independent of x and y, called the absolute term, represents some 
straight line, whatever the values of A, B and C. For if A, B and 
C have finite values, it may be reduced to either of the forms 
y = mx -|- 6, or a; = ny -f- a; if A — 0, that is, if the term con- 
taining x is wanting, to the form y = b ; if B = 0, to the form 
x = a ■; if C = 0, to the form y = mx. This equation includes 
all classes of straight lines, and is, therefore, the general equation 
of the straight line. 

Examples. — Reduce 2x — 3y -j- 5 — to the forms y = mx -j- b 
and x = ny -j- d; x -j- y = 3 to both forms. 

Reduce 3x = 2y to the form y= mx j bx -j- 2 = 0, to the form 
x = a. 

Construction op Equations. 

36. Since an equation of the first degree always represents a 
straight line, to construct its locus it is only necessary to construct 
two of its points and to draw a straight line through them. Thus, 
any two of the points constructed in Art. 17 for the equation 
x = 2y — 3, with the knowledge that it represents a straight line, 
would serve to determine its locus. 

The points usually constructed for this purpose are those in 
which the line cuts the two axes. The distances of these points 
from the origin, or the parts of the axes intercepted between the 
line and the origin, are called the intercepts on the axes. The in- 
tercept on the axis of X is the abscissa of that point of the curve 
which has zero for its ordinate ; it is therefore found by letting 
y — 0, and deducing from the equation the corresponding value 
of x 7 and may be represented by the symbol x . In like manner 
the intercept on the axis of Y is found by letting x = 0, and de- 
riving from the equation the corresponding value of y, and we 
shall denote it by the symbol y . Thus, in the equation x -(- 3# = 3, 
y = gives x = 3, and x = gives y = 1. If we measure off 
these distances respectively on the axis of X and Y, we shall have 
two points through which to draw a straight line, which being 
done we have constructed the equation x -j- 3// — 3. 



CONSTRUCTION OF EQUATIONS. 31 

If one of the terms belonging to the general equation is wanting, 
the line belongs to one of the particular classes of straight lines 
represented by y = b, x = a or y = mx. 

If to the first of these, 5, the constant value of y is also the 
value of y , and the line is drawn parallel to the axis of X, so that 
there is no intercept on that axis. In the second case, x = a, 
and there is no intercept on the axis of Y. In the third case, 
that is, when the absolute term is wanting, both intercepts equal 
zero, because the line passes through the origin, therefore some 
other point besides the origin must be found, thus in constructing 
3x = 2y, we find the point (2, 3) satisfies the equation, and there- 
fore draw a line through this point and the origin. 

Examples. — Find the intercepts and construct the lines, 
x -f y = 3, 2y = 3 — 2x, 4y = 3x, 5x = 7. 

Give in each case the direction ratio, or value of m when reduced 
to the form y = mx -j- b. 

37. General values of the intercepts may be found by using the 
general equation Ax -J- By -\- C = ; thus, 

C _ C 

x = — - andy = — -. 
A B 

When C = 0, both these values becomes zero, as above remarked. 
When A = 0, so that the equation may take the form y = b, x 

takes the form — - , a finite quantity divided by zero, or infinity. 

x Q is generally found by putting y = in the equation j if, how- 
ever, this is done in an equation of the form y = b : for instance, 
y =5, we have the impossible result = 5. This impossible re- 
sult indicates that there is no value of the intercept x , or that the 
line does not cut the axis of X. This must be distinguished from 
the result x = 0, which indicates that the line does cut the axis 
of X at the origin. In the general solution this impossibility of 
the intercept is indicated by the form infinity taken by its value. 
There is another peculiar form which the value of x may take — 

namely, -, when both A = and C = 0. This is called the in- 
determinate form, and indicates that x may take any value whatever ; 
that is, the line meets the axis of X in every point, or coincides with 



32 THE STRAIGHT LINE. 

it. For if A and C are each zero, the equation reduces to ~By = 
or y = 0, which is the equation of the axis of X itself. 

Similar remarks apply to the peculiar forms which may be taken 
by the value of y . 

Intersection of Loci. 

38. When the common point or points of two lines are required, 
their equations being given, we have to find values of x and y, 
which satisfy the two equations at the same time. 

A problem which gives rise to two equations, between x and y, is 
said to be determinate, in distinction from the " indeterminate" 
problems mentioned in Art. 16, because two equations are suihcient 
to determine the point sought ; and the two equations when con- 
sidered in connection are called simultaneous equations ; because, 
arising from the same problem, they are to be satisfied at the same 
time. Thus, if the two conditions of a problem give the equations, 
x = 2y — 3, and x -\- y = 3, we find by elimination that they are 
satisfied simultaneously, by the values, x.= 1, y = 2. When each 
equation is considered by itself, it represents the locus of the con- 
dition expressed in it, Arts. 14 and 20 j and hence the operation of 
combining them to find the values of x and y is the algebraic 
method of finding the intersection of the loci of two conditions, 
which was done graphically, or by actually drawing the loci, in the 
example of Art. 15. 

39. If, as in the above example, the equations are of the first de- 
gree, there is but one solution ; the loci being straight lines which 
intersect in only one point, in this case the point (1, 2). But if 
one of the equations is of the second degree, and the other of the 
first, as x 2 -j- y 2 — J25, and 25 -f- x =7y, we shall have two solu- 
tions ; for, substituting x = 7y — 25 in x 2 -f- y 2 = 25, we have 
50/ — 350?/ -f 625 ±= 25, reducing to y 2 — 7y -f 12 = 0, an 
equation for y of second degree, giving y = 4, or y — 3. The cor- 
responding values of x, in x = 7y — 25 are x — 3 and x = — 4 : 
in fact, in this case, the loci are a circle and a straight line, which 
intersect in the two points, (3, 4) and ( — 4, 3). Both results may be 
verified by substitution in each of the equations. 

Examples. — Find the intersection of 2y = 3.r -j- 7 and 
,y = 5 — \x) of sc a -fy* = ip0 and .z = 8; of x 2 + # 2 = 10 and 



INTERSECTION OF LOCI. 33 

x = 3y ; of x = y and x = — y ; of y — mx — 6 = and my -f- 
x — a = 0. Verify in each ease. 

' 40. When the equations are, in reality, contradictory, as x -f- y = 1 , 
x -{- y = 3, no solution exists ; the equations cannot be solved sim- 
ultaneously, and the problem, supposed to give rise to them both, is 
impossible. The loci of such equations do not intersect at all ; in 
the example, they are parallel straight lines. If we attempt to 
eliminate x between these equations, we have the impossible result, 
= 2. When one of the equations is of second degree, the impos- 
sibility of solution, or the fact that the loci do not intersect, is shown 
by the occurrence of imaginary quantities in the values of x and y. 
For instance, if the equations are x 2 -j- y 2 = 25 and x =y — 8, 
eliminating x we have y 2 — 8 y = — 191, completing the square 
(y — 4) 2 = — 3 h hence y = 4 ± l/^3j. 

To assist himself in understanding this subject, the student may 
construct the loci, in the several examples given, both of intersec- 
tion and non-intersection. 

41. If there are points of intersection, and we combine the equa- 
tions by addition or subtraction, we shall have an equation true of 
these points, whether we eliminate one of the unknown quantities 
or not. For the equations being both true of the common point or 
points, so also must be any equation derived from them. For 
instance, in our first illustration, if we subtract x = 2y — 3 from 
x -\- y = 3, member from member, we eliminate x ; the result is 
y = .6 — 2y, giving at once the value of the ordinate, y = 2. But 
if we add them, we obtain 2x -j- y = 2y or 2x=y, which is true 
of the point (1, 2), which we found to be the point of intersec- 
tion. So also, if we multiply one of the equations through by any 
quantity, before combining ; as 2x -f- 2y = 6, to which if we add 
x = 2y — 3, the result reduces to x = 1, and from which if we 
subtract the same we have as -j- 4y = 9, which is still true of the 
point (1, 2). All of these equations, including x = 1 and y = 2 
(which are the equations of parallels to the axes), are the equations 
of straight lines passing through the point of intersection of the 
original lines. 

This may be generalized for equations of the first degree, 
thus : Let Ax + By + C = and Mx -f B r y -j- C = be the 

B* 



34 THE STRAIGHT LINE. 

equations of any two lines, and k* any number positive or negative, 
then 

Ax -f- By -f C + k ( A'x + B'y + C) = 0, 

is the equation of a straight line passing through their intersection, 
whatever the value of k\ for it is evidently satisfied by those 
values of x and y, which make both the expression, Ax -j- By -j- C 
and A!x -|- B'y ~\- C equal to zero, and it represents a straight line 
because it is of the first degree. 

Examples. — G-ive the equations of a number of lines passing 
through the intersection of 2y = 3x — 7 and x -\-y = 9. 

Form the general equation of the straight line passing through 
this point, and give to k such values as to eliminate, successively, 
x and y. 

42. General values of the co-ordinates of intersection may be 
found by giving k, in the above general equation, successively, such 

values as to eliminate y and x, thus making k = (or making 

the coefficients of y the same, and subtracting), to obtain the value 

of x ; and making k = — — , for y, we have 
A. 

BC — CB' . A'C — C'A 

x = , and y = , 

AB'-BA'' y AB' — BA'' 

for the general values of the co-ordinates of the point common to 
Ax -L- By -(- C = and A'x -j- B'y -f- C = 0, as may be verified 
by substituting these values for x and y in the two equations. 

These values will both at the same time take the value infinity, 
when their common denominator is zero, that is, when AB' = BA', 
or when A :B : : A' :B'. This indicates that the lines do not inter- 
sect, when the co-efficients of x and y in the two equations are pro- 
portional, as in the equations 2x -f- Sy -j- 5 = and 4x -j- 6y -j- 
8 = 0. When this is the case, the general equation above, which 
may be called the equation of combination of the two lines, repre- 
sents a series of parallel lines instead of a series of lines passing 

* The first letters of the alphabet a, b, etc., are used to denote lines or 
distances, as well as x, y, a/, y', etc. ; the letters k, I, m and n are used to 
denote abstract numbers or ratios. The CMpitals A, B, etc., are used to 
denote the coefficients in the general equations. 



INTERSECTION OF LOCI. 35 

through a common point ; as 2x -\- 3y -f- 5 -f- k (4a; -f- 6y -f- 8) = , 
for, being reduced to the form y = mx -f- b, its direction ratio is 

2-\-4k 2 

— = — -, whatever the value of k. 

3 + 6& 3 

Because in the case of parallel lines the general values of the co- 
ordinates of intersection become infinite, parallel lines are said to 
meet in a 'point at infinity. Every equation of the first degree may 
be considered as satisfied by infinite values of x and y, these infinite 
values having a certain ratio to each other, but a finite quantity 
having no ratio to one of them. For example, y = 2x -f- 1 is 
satisfied by infinite values of the variables, of which the value of 
y is double that of x ; y = 2x -J- 2 is satisfied by the same infinite 
values, or passes through the same point at infinity.* The series of 
parallel lines represented by 2x -\- 3y -j- 5 -j- k (4a; -f- 6y -j- 8) = 
may be considered as passing through a common point at infinity ; 
and in general 

Ax + By -f C -f k (A'x -f B'y -f C) == 

represents a series of straight lines passing through a common point, 
even when the given lines are parallel. 

43. If the lines whose equations are thus combined are parallel, 
a certain value of k will cause the terms containing x and y to dis- 
appear at once from the equation. In the numerical example above, 
this value is 7c = — J, which reduces the equation to 1 = 0. For 
every value of k, except this, the equation is of the form A a; -j- 
~By -j- C = 0, and therefore represents a straight line. But for this 
particular value of &, it takes the form C = 0, a constant equals zero, 
which is impossible. jSow the various equations, produced by giv- 
ing different values to k, represent lines passing through a certain 
" point at infinity j" hence the impossible equation of the form 
C — 0, which, strictly speaking, does not represent a line, because 
it can be satisfied by no assignable points, is called the equation of 
the line at infinity, that is, a line all of whose points are at infinity. 

* It must be remembered, however, that a " point at infinity" has no 
position, because no assignable co-ordinates. Its only definite property 
is the ratio conceived to exist between its co-ordinates. All expressions 
involving the idea of infinity are illogical, but convenient as adapting the 
language of the general case (intersection) to the special case (parallelism). 



36 THE STRAIGHT LINE. 

If in the equations Ax -|- ~By -(- C = 0, and A!x -f- B'y -j- 
C = 0, A : A' : : B : B' : : C : C ) that is, if all the corresponding terms 
have the same ratio, the values of both co-ordinates of intersection take 

the form -, showing that the lines meet in every point, or coincide. 

For instance, the equations 2x -j- 3y -|- 5 = and 4x -]- Qy -f- 10 = 
represent the same line, since they are in fact the same equation, 
one being derived from the other by multiplying through by the 
same number. 

Arbitrary Constants. 

44. When the equation of a straight line is in the general form, 
Ax -J- By -f- C = 0, as 2x -j- 3y -j- 5 = 0, it contains three con- 
stants ; and the position of the line depends, not on their absolute 
values, but on their ratios. 

By dividing through by one of the constants, the equation may 
be reduced to a form in which there are but two constants, whose 
values determine the position of the line. Two such forms have 
already been given, y = mx -f- b, and x = ny-\-a; the above 
equation reduced to the first of these forms, is y = — -|x — 1-|; 
reduced to the second, it is x = — fy — 2-1- . 

45. The two constants, occurring in one of these algebraic forms 
are called arbitrary constants, because values may be given them at 
pleasure. The arbitrary constants in y = mx -j- b, are m and b, 
representing the direction ratio of the line, and its intercept on the 
axis of Y. If we wish to make the line fulfil certain conditions, 
we have to determine suitable values for the arbitrary constants. 
Thus, if we wish the equation of a line parallel to 2x -f- 3y -f- 5 === 0, 
we must give to m, in y = mx -j- 6, the same value that it has in the 
given line; namely, — -J, because the line is to have the same 
direction. This gives y= — &x-\-b for the equation; h is still 
arbitrary ; for, whatever its value, the line fulfils the required con- 
dition of being parallel to 2x -j- Sy -j- 5 = 0, or y = — |-x — 1-|. 
b may now be determined so as to make the line fulfil some other 
condition, for instance, that it be twice as far distant from the 
origin ; in which case b must be twice as great as in the given line, 
giving y = — -J-x — 3-J- or 2x -\- 3y -j- 10 = 0. This last result 
might have been obtained by simply doubling the absolute term in 



ARBITRARY CONSTANTS. 37 

the given equation, which it is easy to see has the effect of doubling 
the intercepts. 

Examples. — G-ive the equations of lines parallel to x -(- 2y = 5. 
1st, making the intercept 1 on the axis of Y; 2d, twice as far from 
the origin ; 3d, passing through the origin ; 4th, making the inter- 
cept 3 on the axis of X. 

For the last case assume the form x = ny -\- a, and determine n 
to be the same as in the given line, and a = 3. 

46. When the axes are rectangular, the direction ratio, m, is a 
simple trigonometric function of the inclination of the line to the 

PR 

axis of X ; for PRO, fig. Art. 30, being a right angle, m == = 

OR 

tan POR, the tangent of the line's inclination. In the figure of 

P'R 

Art. 32, m == = tan P'OR, which is the same as the inclina- 

OR 

tion of PB to the axis of X. n, which is the reciprocal of m, is the 
cotangent of the inclination to the same axis. Hence, when the 
axes are rectangular, we can find the inclination of a given line 
to the axis of X by means of the trigonometric tables. Thus 
y = 2x-]-3 makes an angle with the axis of X, whose tangent is 2, 
because m = 2. By the use of 5-place tables, the logarithm of 2 is 
0.30103, and the angle corresponding to this as a logarithmic tan- 
gent is 63° 26' 6", which is therefore the inclination of this line 
to the axis of X. The inclination of y = x -J- 2 in which m = 1 
is 45°, because the tangent of 45° is unity. If m is negative, the 
angle is obtuse, or in the second quadrant, as in the line 2x -j- Sy -f- 
5 = 0, where m = — f = — 0.66667, and the angle will be found 
to be 146° 18' 36". 

Examples. — The axes being rectangular, what is the inclina- 
tion of y = 3x — 2 to the axis of X? of x — 2y -f 3 = ? 
of x -f- y = ? the mutual inclination or difference of inclinations 
of x — Sy -f- 5 = and y = 3x -f 2 ? 

47. If the equation of a line making a given angle with the axis 
of X be required, we may assume y = mx -f- b and compute the 
value of m from the given angle. If it is to make a given angle 
with a given line, for instance, 30° with the line y = 2x -j- 3, we 
ascertain the inclination of the given line, as above, to be 63° 26' 
6", and add or subtract 30°, according as we require the inclina- 

4 



38 THE STRAIGHT LINE. 

tion of the new line to exceed or fall short of that of the given 
one. Suppose it required to exceed it by 30°; then the inclination 
is 93° 26' 6", whose natural tangent is — 16.66. Hence the 
equation is approximately y = — 16.66 x -j- b ) in which b is still 
arbitary, as explained in Art. 45.* 

48. If the line is to be perpendicular to the given line, it is not 
necessary to compute its inclination ; for suppose a to represent the 
inclination of the given line, then tan a is known, and the value 
of m required is tan (90° -f- «). Now, tan (90° -{- a) = — tan 

(90° — a) = — cot a = , because 90° — a is the supple- 

v tan a rr 

ment of 90° -f- a and the complement of a. Hence for a perpen- 
dicular, line, take for m the negative of the reciprocal of m in the 
given line ; thus given y = 3x — 2, in which m = 3, for a per- 
pendicular line we must have m = — -J-, giving the equation 
y = — \x -f- b. The condition that two given lines should be 
perpendicular is, that the values of m should be of opposite signs 
and reciprocals, thus y = x -\- 3 and y = 3 — x are perpendicular. 

Examples. — Give the equations of a number of lines perpen- 
dicular to 3x -f- y = 5 ; of a line passing through the origin and 
perpendicular to 7y — 6x -f- 3 = 0. 

Of the lines 3x -f 2y -f 1 = 0, 3x — 2y -f 1 = 0, 2x + 3y -f 
1 = 0, 2x — 3j/ -\~ 1 = 0, which are perpendicular ? 

It must be remembered that the results of the last three Articles 
do not apply to oblique co-ordinates, in which case m is not a sim- 
ple function of the inclination of the line to either axis, but de- 
pends upon its inclination to both axes, and hence upon the angle 
.between them.f 

* When the angle between two given lines is required, we may with 
advantage use the formula for the tangent of the difference of two angles, 

tan (0 — Q') = ; thus, given y = 2x — 3 and y = x -f- 2 

1 + tan 9 tan 9' 
tan = 2, tan 9' = 1, hence tan (9 — 9) = i When a line making given 
angle with given line is required the method in the text is the simplest in 
computation. 

t In general, let a = POR, figure of Art, 30, or the line's inclination to 
the axis of X, = POY = OPR, and u = YOX, then by trigonometry 

PR sin POR sin a , , „ TT «in « 

m = — = == , and a -\- p = o>. Hence m 



OR sin OPR sin/? sin(a) — a) 



FORMS OF THE EQUATION OF THE STRAIGHT LINE. 39 

Forms of the Equation of the Straight Line. 

49. In finding the algebraic equation of the straight line, we 
may employ any constants whose values determine the position of 
the line. In order that the equation should be capable of repre- 
senting any line, there must be at least two constants j an equation 
containing but one can only represent a particular class of lines ; 
as y = mx, lines passing through the origin j x ■■= a, lines parallel 
to the axis of Y. In y = mx + b, the two constants are the ab- 
stract number m, or direction 
ratio and the length, or number 
of linear units b, which is the in- 
tercept on the axis of Y. In 
x == ny -{- a, they are the recip- 
rocal of the direction ratio and 
the intercept on the axis of X. 

50. Let us now find the equa- 
tion of the straight line in terms 
of its two intercepts. In con- 
formity with the notation of the 

previous equations let a represent the intercept on the axis of X, 
and b on that of Y ; then drawing the ordinate of any point, P, we 

have, by similar triangles, b : y : : a : a — x. ori = === 1 — -. 

' J ° J ' b a a 

Hence - + y - = 1 

a^ 

is the relation existing between the constants a and b and the 

co-ordinates of any point of the line. This is the equation of a 

line in terms of its intercepts ; for let y = 0, and we have x = a; 

let x = 0, and we have y =b. If x be taken negative, in the 

V 
figure, where a and b are positive, y is greater than b and - ^> 1, 

b 

x x 

but - is then negative ; so that the algebraic sum of the quotients - 

a a 

y 
and - is still unity. If a is negative, as it would be for the line 
b 

which renders the process of finding a from m much more difficult. When 
o = 90° this value of m reduces to tan a. 




40 



THE STKAIGHT LINE. 



BP, figure Art. 32, - is negative and y > &,for positive values of x. 

a 
By means of this formula we can give at once the equation of a 
line making given intercepts ; thus, required that x = 7 and y = 5, 

1, or hx -f- 7y == 35 ; required 



the equation of the line is - 4- - 

H 7^5 



x = — 2, y = 2 the equation 



ory 



= 2. 



If the line is parallel to one axis, for instance the axis of X, the 

x 
intercept on that axis is infinite, and the fraction - = 0, the equa- 

a 

V 
tion therefore reduces to - = 1 or y = b, as before found. If the 

b ' • ' 

line passes through the origin, the intercepts are both zero, and the 
fractions become infinite, so that the equation of this class of lines 

x II 

cannot be put in the form --)-■-== 1 
a ^ 

Examples. — Give the equations of lines making the intercepts 
a* = 1, yo.= — -2 ; x Q = — 5, y = 10 ; x = — 1, y = cc, 
etc., etc. 

In each case reduce the line to the form y == mx -f- b, and give 
its direction ratio. 

x u 
Find a general expression for the direction ratio of — {- - r= 1. 

a ft 

Ans. m = . 

a 

51. When the axes are rectangular, an especially convenient form 
of the equation of the straight line is that in which the constants 
are the length of the perpen- 
dicular from the origin on the 
line, and the inclination of this ^^ 

perpendicular to the axis of X. 
In the figure, p represents the 
perpendicular and a its inclina- 
tion and a and b the intercepts. 
If now we find values for a and 
b, in terms of p and a, and sub- 




stitute them in 



1, we shall have the required equation. 



FORMS OF THE EQUATION OF THE STRAIGHT LINE. 41 

P 
By the definitions of the trigonometric functions _ = cos a and 

a 

_ = sin a. Hence the equation, 

x cos a -J- y sin a = p. 

In this form the arbitrary constants are a distance p, and angle a, 
whose functions cos a and sin a are ratios or abstract numbers. In 
the figure, a is in the first quadrant or < 90°. If a were in the 
second quadrant, it is evident the line would make a positive inter- 
cept on the axis of Y, and a negative one on the axis of X, in which 
case, cos a would be negative and sin a positive. If a were in the 
third quadrant, both cos a and sin a, and both intercepts, would be 
negative ) if in the fourth quadrant, cos a is positive and sin a nega- 
tive, hence x is positive and y Q negative. Hence we see that with 
a proper value of a, admitting angles in all the four quadrants, the 
equation of a line in any position may be expressed in this form, the 
value of p being considered always positive. Thus, if we require 
the equation of a line seven units distant from the origin, we have 
x cos a -\-y sin a = 7, which by giving different values to a, may 
be made the equation of any straight line at that distance from the 
origin. For instance, a = 0° gives the equation x = 7, a = 90° 
gives y = 7,a = 180° gives — x = 7 or x = — 7, a = 270° gives 
— y = 7 or y = — 7, a = 45° gives V \ x -j- V \ y = 7, or 
x -f y == 7l/2, a = 135° gives — V\ x -f V\ y — 7, or y — 
x = 7l/2. 

52. The equation of any straight line can be written in this form, 
in which the absolute term is the perpendicular distance of the line 
from the origin. To put an equation in this form, it is not 
necessary to find the value of the angle a, but only to make the 
coefficients of x and y in the equation the sine and cosine of some 
angle. Now the sum of the squares of the sine and cosine of any 
angle is unity. If then the coefficients have this property, the 
equation is in the form x cos a -\- y sin a =p, though the value of 
a is not stated ; thus &x — 1# = 3 is in the above form. If the 
sum of the squares of the coefficients is not unity, we must divide the 
equation through by the square root of this sum : thus, given 
12^ — 5x —J— 26 = 0, we find the sum of the squares to be 169; we 
therefore divide the equation through by |/l69 or 13, which gives 



i2 THE STRAIGHT LINE. 

■J-Jy — -f^x — |— 2 = 0, or -f^x — \^y = 2, which is in the required 
form, and shows that the perpendicular from the origin on the line 
is two units in length. The inclination of this perpendicular is the 
angle whose cosine is T 5 g-, and whose sine is — -l~|, an angle evidently 
in the fourth quadrant, and whose value can be found from the 
trigonometric tables. Usually the divisor will be a surd; as for 

x y 

x-\-y = 2, which reduced is ~~ -j- ~~r =1/2, the perpendicular 

1/2 V _& 

being l/2. 

Examples. — The axes being rectangular, what is the distance 
of the line ?>x — y = 4 from the origin ? of the line 4x — 3y = 10 ? 
of the line 7x = 21 ? etc., etc. 

53. A general expression for the distance of a line from the 
origin may be found by reducing the general equation Ax -\- By -f- 
C = to this form, which gives 

A , B — C 

-y 



VA 2 + B 2 l/A 2 -f B 2 l/A 2 + B 2 ' 

C 



the distance from the origin is therefore numerically 

l/A 2 + B 2 ' 
the absolute term divided by the square root of the sum of the 
squares of the coefficients. We cannot tell whether we ought to 
change the signs of the above equation throughout, unless we know 
the sign of C. 

The equations x — a and y = b are already in this form j accord- 
ingly a and b are perpendiculars from the origin upon these lines, 
when the axes are rectangular. 

54. If in the equation x cos a -j- y sin a =p, we make p = 0, 
we have a parallel line passing through the origin; if we make p 
negative, the effect is the same as if we had changed the sign of 
both cos a and sin a, in the first member, which is equivalent to 
replacing a by a -j- 180°, or measuring the perpendicular in a 
directly opposite direction. All lines having the same value of a, 
or values differing by 180°, are parallel, but to ascertain in which 
direction the perpendicular is measured, we must make the second 
member of the equation positive, and find the corresponding value 
of a. Thusffx — ^# = 1, ffx — fV<y = 0, and 1.2 X — ^y = — l, 



EQUATIONS OF CONDITION. 43 

are parallel ; but in the first case a = 337° 22' 48", in the fourth 
quadrant, because cos a is positive, and sin a is negative, and in the 
third case a = 157° 22' 48" r in the second quadrant; for changing 
signs throughout to make the second member positive, we see that 
cos a is negative, and sin a positive. In the second equation it is 
immaterial which value we give to a. 

Examples. — Give the value of a, in 3x — y = 4, 3x=y, 
3x — y = — 4; in x -\- y = 1, x -\- y — — 1; in 7x = 21, 
7x = — 21, etc., etc. 

Equations of Condition. 

55. In preceding articles we have found several algebraic forms 
of the equation of the straight line, in which, by directly giving 
proper values to the constants, we produce the equations of lines 
fulfilling certain conditions : we shall now give a general method, 
by which the constants in an equation may be determined so as to 
make the line pass through given points. Suppose, for instance, it 
is required to find the equation of a straight line passing through 
the point (7, 5). Assume the equation y = mx-\-b, which we 
know to be the equation of a straight line, whatever the values of 
m and b, the proper values of these constants being, however, as yet 
unknown. Now, by the condition, the point (7, 5) must satisfy the 
equation ; hence 5 — 7m -j- b. Whenever m and b are so related 
that this equation is true, the given point satisfies the equation ; or 
the equation represents a line passing through the point. It there- 
fore imposes a condition upon m and 6, equivalent to the condition 
proposed in the problem. As there are two unknown quantities, m 
and b, to be determined, we can satisfy two such equations of con- 
dition. Thus, let the line be required also to pass through the 
point (2, — 7) ; the equation expressing this condition is — 7 == 
2m -j- b. Solving these two equations of condition, we have m = 2-|, 
b = — 11-J, which are the only values of m and b which will 
satisfy them both. These are therefore the proper values of m and 
b in the assumed equation y = mx -j- b ; substituting them, we 
have y = 2%x — 11|-, or 5y = 12x — 59. The result may now be 
verified by substituting the co-ordinates of the given points for x 
and y, and showing that each point satisfies the equation. 

Examples. — Find the equation of the straight line passing 



44 THE STRAIGHT LINE. 

through ( — 1, — 2) and (0, — 3); through the origin and 
(1, — 1,); etc., etc. 

Find the equation of the line making the intercepts x = 6, 
y = 8, as the line passing through (6, 0) and (0, 8). 

56. We can thus find the equation of a straight line fulfilling two 
conditions, or passing through two given points, because there are 
two arbitrary constants to be determined, which allows us to satisfy 
two equations of condition and no more. If we should assume the 
form Ax -j- By -j- C ■== 0, and attempt to satisfy three equations of 
condition, we should find A = 0, B = and C = 0. In general 
we can satisfy as many equations of condition, and hence make a 
line pass through as many points, as there are constants in the 
assumed equation, but we must assume the equation in such a form 
that one of the terms contains no constant, otherwise every equa- 
tion of condition will be satisfied by making all the constants zero. 
Thus, Ax 2 -J- Dx -\- E y -j- F = is the general equation of a cer- 
tain class of curves. If, however, we wish to find that curve of the 
class which passes through certain given points, we must assume the 
form x 2 -|- Dx -f- Ey -f- F — 0, and then it is evident, that three 
equations of condition may be satisfied, or the curve may be made 
to pass through three given points. 

Formulae for Straight Lines. 

57. The two constants in either of the algebraic forms, y = mx-\- 6, 

x II 

x = ny -f- a, — |- - = 1, and x cos a -j- y sin a ==p, are quantities 
a i) 

having particular values for a given straight line, which values may 
be ascertained by reducing the equation to the proper form : thus 
we reduce the equation to the form y — mx -f- b to ascertain the 
line's direction ratio; to the form x cos a -\- y sin a =jp to find its 
distance from the origin. 

These equations are also formulas, by which the equation of a 
line is expressed in terms of certain constants whose values may be 
given, as in the examples of Art. 50, or found by an algebraic pro- 
cess as in Art. 55. 

58. The equation of the straight line may also be expressed in 
terms of the co-ordinates of known points on the line, and then we 
shall have formulae for lines passing through given points. This 



FORMULA FOR STRAIGHT LINES. 45 

may be done by means of the method of equations of condition, ex- 
plained in Art. 55. 

Suppose, in the first place, the co-ordinates of P', a point of the 
line, to be known, or the line required to pass through a known 
point (V, y). The equation y = mx -f- b being assumed, we have, 
because P' is a point of the line, the equation of condition, 

y = mx' -j- b. 

By means of this we can eliminate from the equation of the line 
one of the arbitrary constants, introducing in its stead the known 
quantities x' and y. Thus, substituting for b in y = mx -\- b its 
value, b =y' — mx', from the equation of condition, we have 
y = mx -J- y — mx*, or 

y— y = m(x — a/), 

in which the arbitrary constant m is retained. 

This is a formula for a straight line passing through a given 
point. It still contains an arbitrary constant, because there is a 
variety of lines passing through a given point. In fact, whatever 
the value of m, the line evidently passes through P', because sub- 
stituting x' and y for x and y both members reduce to zero, inde- 
pendently of the value of m. By means of a proper determination 
of m, the line may be made parallel to a given line, or, the axes 
being rectangular, perpendicular to a given line. Thus, for the 
line passing through (7, 5), substituting y = 5 and x' = 7, we 
have y — 5 = m (x — 7). If the line is also to be parallel to 
2y — x -\- 3 = or y = \x — |-, we make m = 1, giving y — 
5 = 1 (x — 7), or 2y = x -f- 3. If it is to be perpendicular to 
this line, we make m = — 2, giving y — 5 == — 2 (x — 7), or 
y = 19 — 2x. The results are verified by showing that they are 
satisfied by the values x = 7, y = 5. 

Examples. — Find the equations of lines passing through 
( — 1, 2): 1st, parallel; 2d, perpendicular to 3y -{- 2x = 5 : 
through (0, 6) parallel to y -f- x = 0, etc., etc. 

What is the general formula for a line passing through P' and 
perpendicular to y = mx -\- b? 

Am. y — if =■ — — Qz — #0- 

59. The formula y — y = m (x — x') is a more general one, 



46 THE STRAIGHT LINE. 

including the previous ones in which m occurs ; for, make x' = 
and y = 0, and it reduces to y = mx, the equation of a line pass- 
ing through the origin ; make x' = and y' = b, the co-ordinates 
of B in the figure of Art. 32, and it reduces to y — b = mx, or 
y = mx -f- b, the equation of a line making the intercept b on that 
axis. On the other hand, it is a special case of the still more 
general formula of Art. 41, formed by combining the equations of 
two lines in their general form, 

A* + By + C + k (A'jc + B'y+ C') = 0. 

For x = x f and y = y' are the equations to two straight lines pass- 
ing through P' ; namely, the lines parallel to the axes. Putting 
them in the forms x — x' = 0, y — y'= 0, and combining, we 
have y — y' = — k (x — a/), or, writing the letter m instead of 
— k for the arbitrary constant, y — 7/ = m (x — a/). So y = mx 
may be derived by combining the equations of the axes, y = and 
x = 0, and is therefore the general equation of a line passing 
through their intersection, the origin. 

60. The general equation of combination above, as well as the 
present formula, which is a special case of it, contains one arbitrary 
constant, because it represents a line fulfilling one condition — * 
namely, that of passing through a particular point ; but not deter- 
mined by that condition, inasmuch as a straight line can be made 
to fulfil two conditions. That constant can now be determined so 
as to make the line fulfil any other condition required. For ex- 
ample, required the equation of a line passing through the inter- 
section of 2y = 3x — 7 and x -|- y = 9, and parallel to y = 2x -J- 3. 
We may use the method of combining the equations to avoid the 
necessity of finding the intersection. Thus, multiplying the mem- 
bers of x -j- y = 9 by k, and adding to those of 2y = 3x — 7 (it 
is not essential to transpose all the terms to one member), we have 
2y -f- kx -j- ky = 3x — 7 -J- 9k ; reducing this to the form y = 

3 fa 9^. 7 

mx -j- 5, it becomes y = x -j . Now, to be parallel 

2 -j- k 2 -\- k 

3 fc 

to v = 2x 4- 3 we must have the direction ratio = 2. 

J ^ 2 + k 

This determines the value of k ; for clearing of fractions, 3 — k = 
4 -j- 2k, or k = — J. This value we may now substitute in the 



FORMULAE FOR STRAIGHT LINES. 47 

original equation, or in the value of b in the reduced equation, 

which gives b = — — 6, hence the equation required is 

y = 2x — 6. k may also readily be determined, so as to make the 
line pass through a given point. Suppose the above line 2y -f- 
kx -(- ky = 3x — 7 -f- 9k required to pass through (1, 2). The 
equation of condition is 4 -}- h -j- 2& = 3 — 7-j- 9&, hence & = 1-J, 
substituting which, we have for the equation of the line 2y -j- 1-J- x 
-f Hy=3x — 7-f 12, or 1 1 Py-=.#«;-h5, or 2y = a; -{- 3. This 
is verified for this last condition by substituting 1 and 2 for x and 
y, and the first condition — that it pass through the intersection 
of 2y = 3x — 7 and y -\- y = 9 — may be verified by finding that 
intersection, which is (5, 4), and substituting in 2y = x -j- 3. The 
equation y -j- 2x — 6 is also satisfied by (5, 4), hence both the 
lines found pass through the intersection of the given lines. 

Examples. — G-ive the equation of the lines passing through the 
intersection of 2y = 3 — 4x and 4 — y = 5x : 1st, parallel ) 2d, 
perpendicular to 3y = 2x -f- 6 ; through the intersection of x = 5 
and x — y = 9 : 1st, parallel to x -f- y = ; 2d, passing through 
(2, 1) ; of 2y -J- x = 5, and 3 — y = x : 1st, parallel to as — y = 
10 ; 2d, passing through ( — 3, 5), and verify. 

61. Every equation of first degree containing only one arbitrary con- 
stant may be regarded as a case of Ax -j- 3y -\- C -j- k (A!x -j- ~B r y -f- 
C) == 0, and therefore represents a series of lines passing through a 
single point. This point is the intersection of two straight lines 
whose equations consist, one of those terms which do not contain 
the arbitrary constant, and the other, of those which do. Thus 
x -J- 2y x x -f- y x y =25, where y x is an arbitrary constant, is the 
equation of a line passing through the intersection of x = 25 and 
2x -f- y = 0, that is, through the point (25, — 50) ; for, in x — 25 -f 
k (2x -j- y) = 0, we have only to put y x in place of k to produce 
the given equation. Therefore whatever the value of y x the line 
passes through this point. This is verified by showing that x = 25, 
y = — 50 satisfies the equation independently of y v 

Examples. — Through what point does the line l(x — y) -j- 3x -f- 
2y = 2 -j- I, necessarily pass ? 

Ans. The intersection of 3x -j- 2y = 2 with x — y — 1, which 
is the point (|-, — 1). 



48 THE STRAIGHT LINE. 

Verify this result for a number of assumed values of I, as when 
1=1, when I == 2, etc. 

Find the point common to the lines represented by 2cx — 
(y — c) = 3;(y + a0 + 4(*-l). 

62. The most ready way of finding the co-ordinates of the point 
sought in the above examples, is to give the arbitrary constants 
such values as to eliminate successively x and y from the equation ; 
thus finding the equations of two lines of the series — namely, those 
parallel to the axes, or of the forms x = a and y = b. 

For instance, in I (x — y) -j- 3x -f- 2y = 2 -f- I, we eliminate y 
by making I = 2, which gives 5x = 4 or x = ^, the equation of a 
line parallel to the axis of Y passing through the required point, 
hence -| is the abscissa of that point. Similarly, I == — 3 gives an 
equation of the form y = b, 5y = — 1 or y = — ^-. This is in 
fact the same process as that used in Art. 42 to find general values 
of the co-ordinates of intersection. By giving I in the example the 
value — 2, we shall have an equation of the form y = ax, because 
1= — 2 makes the absolute term disappear. The result x -j- Ay = 
is therefore the equation of the line of the series passing through 

C 
the origin. In general, giving k the value , we have for the 

Vj 

equation of the line passing through the intersection of Ax -{- 
By -f C = with A'x -f- B'y -fC' = 0, and the origin, 

C (Ax + By + C) — C (A'x + B'y -f C) = 0. 

Q 

The value h = might have been found by the condition 

that the origin should satisfy the equation. 

63. If the two equations into which we decompose an equation 
with one arbitrary constant, are the equations of parallel lines, as in 
the example, I (x — y~) -\- 3x — 3y = 10, the point common to the 
system of lines is at infinity ; that is, the given equation represents 
a series of parallel lines; in the example, all parallel to x — y — 0. 
So too, if one of the equations is of the form C = (a constant 
equal zero), the impossible equation, which in Art. 43 was in- 
terpreted as the equation of the line at infinity; as I (x — y) = 1, 
which represents the same series of parallel lines, all parallel lines 
meeting the line at infinity in the same point. The equation 



FORMULAE FOR STRAIGHT LINES. 49 

y 1=3 mx -f- b, when b alone is considered as an arbitrary constant, 
represents a series of lines passing through a common point at 
infinity, the point in which y = mx meets the line at infinity. 

Examples. — What series of lines is represented by l(x — 2) -{- 
l(y-x) = (l+2)y + 3x-l? 

What, by the general equation Ax -f- By + C = when B and 
C are fixed, and A is an arbitrary constant. 

Ans. A series of straight lines passing through the intersection 

of x == and By -f- C = 0, or the point (0, ]. 

What, when B alone is arbitrary ? and what when C alone is 
arbitrary ? 

64. The equation of a line passing through a given point, 
y — y f = m(x — a/), contains one arbitrary constant; by means 
of an equation of condition this constant may be determined so as 
to make the line pass through another given point. Let P" be this 
second point, then 

y" — V 

y"-*J = m (x" - x'), or m = #— * 

is the condition that P" should be a point of the line ; substituting 
this value of m in y — i/ = m (x — a/), we have 

X — X 

which is the formula for the straight line passing through P' and 

P". In making use of this formula, the co-ordinates of one of the 

given points are substituted for x' and y, and those of the other 

for x" and y" : thus required the line passing through ( — 1, 2) and 

3 

(3, — 1); calling the first point P', we have y — 2 == (x -f- 1), 

which reduces to 4y -f- 3x = 5. The result is verified by showing 

that each given point satisfies the equation. If the second point 

had been taken as P' and ( — 1,2) as P", we should have y -\- 1 — 
3 

(x — 3), reducing also to 4ty -f- 3x = 5. 

In applying this formula it is best first to find the value of the 

y" — i/ 

direction ratio — , Two special cases may occur : when the 

x" — x' 
5 C 



50 THE STRAIGHT LINE. 

ordinates of the two points are the same, the direction ratio be- 
comes zero and the equation reduces to y — y' = or y = y' • and 
when the abscissas are equal, the direction ratio is infinite and the 
equation of the line is x = x' . Thus, the line passing through 
(1, 2) and ( — 3, 2) isy = 2; that passing through (1, 2) and 
(1, — 1) is a; = l'. In the latter case the formula y — y f — 

v" — v' 

— (a; — x') may be regarded as expressing that y is inde- 

x" — x' 

y" _ y> 

terminate, for x — x' always equals zero, when — is infinite, 

x" — x' 

and X oo, like -, is indeterminate in value. 
' 

Examples. — G-ive the equation of the line joining ( — 2, 0) and 
(1, — 1) : passing through (6, 3) and (2, — 1), etc., etc. 

The equations of the sides of the triangle whose vertices are 
(1, 2) (2, 3) and (3, 1). 

The sides of the triangle whose vertices are (6, 2), ( — 1 — 8) 
and (— 3, 0). 

Find the equation of the line joining (a, 0) and (0, b). 

Show that the point found in Art. 11 satisfies the equation of 
F P". 

65. As the formula for lines passing through one or two given 
points are very important, we give also, the method of deriving 
them geometrically, that is, 

directly from a figure. Let » p 

P' and P" be the given 1 5 

1 P' ^^iR' 

points, and P any point of 
the line. Draw the ordi- 
nates of these three points, 
and parallels to the axis of " 
X through F and P": then 
PR=y_y and FR = 
x — x'. It is evident that 
wherever on the line P be 
taken, PR and P'R will have the same ratio, which is the direction 

y — if 

ratio of the line, and may be denoted by m, hence — = m, or 

x — x 

y — y' = m (x — a?'). 






DEMONSTRATION OF GEOMETRICAL THEOREMS. 51 

P"R' y"—y' 
But = — = m, hence 



P'K 



a; — x 
This equation may also be derived from the similar triangles PRP', 
P"B/P', which give the proportion^ — y' :x — x f : : y" — y : x" — x'. 
For a line passing through a given point and the origin, make 

v" 

x' = 0, y = 0, and we have y = — x, or writing x\ y' in place of 

x" 
x'\ y", since there is but one given point, 

y = — x, or xy = yx. 

Thus, the line joining (7, 5) to the origin is 7y = 6x. 

Demonstration of Geometrical Theorems. 

66. The analytic method is well adapted to the demonstration 
of geometrical theorems in which it is proved that certain lines 
pass through a common point. For instance, to prove that the 
three lines joining the vertices of a triangle with the middle points 
of the opposite sides pass through a com- 
mon point. Let ABC be any triangle, 
and MNO the middle points of its sides. 
We simplify the demonstration by taking 
as axes the lines AO and BC ; then we 
have to form the equations of MC and 
BN, and prove that they cut the axis of Y 
— the line AO — in the same point. De- 
note the distance AO by b, and OC by a. 
Then the co-ordinates of the middle point N, between A(0, 6) and 
C(a, 0), are \a and \b ; those of M are — \a and \b. Using the 
formula for a line passing through two points, the equation of BN 
passing through B (x' = — a, y' = 0), and N (x" = |a, /' = lb) 

is y = — h (z + a). To find where this line cuts AO, the axis 
of Y, make x = ; the result is y = ib. Similarly, the equation 
of CM is y = — ~ (x — a), in which also y = }b; hence BN and 




52 THE STRAIGHT LINE. 

CM cut AO in the same point, each cutting off one-third of its 
length. Thus we have not only demonstrated the existence of a 
common point, but found its position on the line AO. 

67. The three lines here proved to meet in a point are called 
the bisectors of the sides of the triangle. By means of the formulae 
of Art. 11 for the middle point of any line, and the formula for a 
line passing through two points, we may form the equations of the 
bisectors for any given triangle, and then find the point where two 
of them intersect, and show that it satisfies the equation of the third. 
Thus, in the triangle whose vertices are (1, 6) ( — 3, 2) and 
(5, — 1), the middle point of the first side is ( — 1,4), the line 
joining this with (5, — 1) is found to be 6y -j- 5x = 19. Simi- 
larly the equations of the other two are 12y = a? —J- 27, and x==l: 
these meet in the point (1, 2 J) and this point satisfies Qy -J- 5x = 19. 
If we let P l5 P 2 , P 3 represent three given points, we might by this 
method find formulae for the bisectors and the co-ordinates of the 
point where they meet, in terms of the vertices of the triangle, and 
give a general proof that the bisectors of any triangle meet in a 
point.* As a method of proof, it would be much less simple than 
the one we have used, which is perfectly general as a demonstra- 
tion, because ABC was any triangle, although for the sake of sim- 
plicity the axes were chosen in a particular way. 

Examples. — Find the bisectors of the triangle (1, 2) (2, 3) and 
( — 3, 1), and show that they meet in a point. 

Find the bisectors of the triangle (6, 2) (—1, —8) and (3, 0), 
and the point in which they meet. 

Prove the theorem, taking two sides of the triangle as axes. 

Find the equations of perpendiculars to the sides of the above 
triangles at their middle points, and show that these lines also meet 
in a point, the axes being supposed rectangular. 

Prove that these perpendiculars in any triangle meet in a point, 
taking as axes one of the sides and its perpendicular. 

* The equation of the bisector of the side P 2 P 3 is (y — y x ) (x 2 -f- x 3 — 
2x x ) =(y 2 + y$ — %i) (% — x 1 ), from which, by interchanging the subscript 

(T I T | o* 
_J _? • 3 > 

J\ ~r 2/2 -r Vz \ j g t j ie common point, for it satisfies the above equation and 

3 / 

by symmetry must satisfy the" others. ■ 



POLAR EQUATION OF STRAIGHT LINE. 53 

Find the equations of perpendiculars drawn from the vertices of 
the above triangles, to their opposite sides, and show that they also 
meet in a point. 

Prove that in any triangle these perpendiculars meet in a point, 
taking one of the sides and its perpendicular as axes. 

Polar Equation of Straight Line. 

68. The polar equation of a straight line is the relation which 
exists between the polar co-ordinates of each of its points. If, how- 
ever, the line passes through the pole, as in the figure of Art. 28, 
is constant for all points of the line, at least, when negative values 
of r are admitted, thus = 100°, = 40°, and in general = a 
where a stands for the line's inclination, represents a line passing 
through the pole. In all other cases is variable, and we must 
find a relation between the co-ordinates of a point of the line, and 
certain constants. Let be the pole, OA the initial line, and PR 
any straight line : let us use 

the same constants as in the \p 

rectangular equation of Art. .,•-"' \ 

51 — namely, p : the length of /' \ 

the perpendicular, OR, from S'\e j£^^\ 

the pole, and a its inclination --^--^^T\ \ 

to the initial line. Draw OP l " J V 

the radius vector of any point V 

of the line; then ORP is a \ 

right-angled triangle, of which \ 

the angle POR is represented 

by — a, r is the hypothenuse, and OR the side adjacent to this 

angle; hence by the definitions of the trigonometrical functions 

cos (0 — a) = _, or' 
r 

r cos (0 — a) =_p- 

69. This is the relation between r, 0, p and a, and hence is the 
equation of the line PR. The constants, p and a of this equation, 
are the polar co-ordinates of R, the point of the line nearest the 
pole ; accordingly if we let = a, (0 — a being zero, whose cosine 
is unity), we have r=p. If we take P, in the figure, below the 
point R. — a is negative, and < 90°. or in the fourth quadrant, but 



54 THE STRAIGHT LINE. 



its cosine is still positive, and r is positive. Since r — 



cos {0 — a) 7 

the smallest value of r corresponds to the greatest value of the cosine, 
which is given by = a • as increases the cosine decreases, and r 
increases, and when = 90°-}- a, cos(# — a) = 0, and r is infinite, 
that is, there is no point of the line in a direction from the pole 90° 
in advance of that of OR. When exceeds this value, r is nega- 
tive, in which case the line, drawn in the direction 0, diverges from 
the line PR, but being produced backward through the pole would 
intersect it. When = 270° -J- a, the value of r is again infinite; 
and when it exceeds this value, r is positive. Thus during a com- 
plete revolution of the radius vector, its extremity P describes the 
line twice ; since values of 0, differing by 180°, evidently give the 
same values of r with opposite signs, and therefore lead to the same 
point. During succeeding revolutions the line is described again, 
as values of differing by 360° give the same values of r. 

70. The constants, p and a, which determine the position of the 
line, being also the polar co-ordinates which determine the position 
of R, r cos (0 — 'e£)=p is a formula for the equation of a line 
whose nearest point is given. Thus, the co-ordinates of the nearest 
point being (7, 100°), the equation is r cos (0 — 100°) = 7. If 
we use the co-ordinates ( — 7, 280°), which denote the same point, 
as constants, we shall have r cos (0 — 280°) == — 7, which is, in 
fact, the same equation, the signs of both members having been 
changed at once. 

Thus we interpret equations in which p is negative ; when such 
is the case, we can make p positive by adding 180° to, or subtract- 
ing it from, the value of a. If p = 0, the equation reduces to 
r cos (0 — a) = 0. This is satisfied by r = 0, and any value of 0, 
that is, by the pole, but if r has any other value than zero 
cos (0 — o) = 0, hence — a = 90° or 270°, that is, = a + 90° 
or = a-j-27O°. Now 0=a is the equation of the straight line 
OR passing through the pole and making the inclination a, and 
either of the above is the equation of a line passing through the 
pole and perpendicular to OR; that is, parallel to PR. Hence 
all lines having the same value of a, or values differing by 180°, 
their equations being in the form r cos (0 — a) == p, are parallel 
to one another and perpendicular to the line 6 = a. 



POLAR EQUATION OF STRAIGHT LINE. 55 

71. By giving the proper value to a, the line may be made to 
have any given direction. If a = 0, it is perpendicular to the 
initial line, and the equation reduces to 

r cos 6 = p, 

in which, if p is positive, the line cuts the initial line on the right 
of the pole, but if p is negative, it cuts the initial line produced to 
the left of the pole. In the latter case we might make p positive 
and a = 180° ; thus r cos 6 = 5 and r cos 6 = — 5. are perpen- 
dicular to the initial line, and the latter may be written r cos (0 — 
180°) = 5. 

If a = 90°, the line is parallel to the initial line, and since 
cos (0 — 90°) == cos (90° — 0) = sin 0, the equation reduces to 

r sin 6 =jp, 

in which, if p is positive, the line is above the initial, line, if p is 
negative, below it ; the proper value of a in the latter case being 
270°. Thus r sin 6 = 3, r sin = — 3, are parallel to the initial 
line. 

Examples. — Give the polar equation of the line whose nearest 
point is (5, 45°) ; that for which a = 60°, p = 3 ; that for which 
a = M°,p = — 3. 

Give the equation of parallels to the initial line, six units above 
it and below it. 

72. Owing to the difficulty of working with trigonometric equa- 
tions, the polar equations are not well adapted to finding the inter- 
sections of lines. The following is one of the simplest class of ex- 
amples : Find the intersection of r cos 6 = 7 and r sin = 5. We 
eliminate r by division and obtain tan = f, from which, by the 
trigonometric tables, we find the value of 6 to be 35° 32' 15". We 
might also take 6 = 215° 32' 15", an angle having the same tan- 
gent as 35° 32' 15", but having a negative sine and cosine j but we 
take 6 in the first quadrant, so that r, as found from r cos 6 = 7 or 
r sin 6 = 5, will be positive. This value will be found to be ap- 
proximately r = 8.6023. If we wish to find only the value of r, 
we may eliminate 6 from the original equations by squaring and 
adding, for cos 2 -f- sin 2 = 1; hence r 2 = 74, r = ± l/74 
according as we take 6 in the first or third quadrant. The inter- 



56 



THE STRAIGHT LINE. 



section of the lines n cos d = 7 and r sin d == 5, is the point 
(l/74, 35° 32' 15") or (— i/74, 215° 32' 15"). 




Distance of a Given Point from a Given Line. 

73. We suppose, now, the axes to be rectangular, and that it is 
required to find the distance of a point given by its co-ordinates, 
from a line given by its equation. Instead of finding the co-ordi- 
nates of the foot of the perpendicular, and using the formula of Art. 
12, we may put the equation in the form x cos a -f- y sin a =p, 
in which p is the value of the perpendicular from the origin. Then, 
evidently, the value of the 
expression x' cos a -\- 1/ sin a 
would be p, if the point P' 
were on the line ) if not, it 
is the perpendicular from 
the origin on a parallel line 
passing through P'. Thus 
in the figure, the construc- 
tion of which is apparent, 
the valufe of x' cos a -)- y ! 
sin a is the length of OR. The difference between this and p is the 
perpendicular required, which denote by p' ; then 

x' cos a -j- y sin a — p = p' 
is the formula for the perpendicular from P' upon the line 
x cos a -j- y sin a — p = 0. 

In fact, the formula is equivalent to an equation of condition, 
expressing that P' is on the line, 

x cos a -}- y sin a =p -\- p\ 

which is the equation of the parallel through P', of which RP' is 
a part. 

74. The dotted lines of the figure are drawn to indicate a geo- 
metric or direct method of proving the formula ; for OR = OS -f- 
P'T ; now by the definitions of trigonometry OS = x' cos a, and 
P'T = y f sin a. Therefore p' = OR — p = x' cos a -j- y sin a — p. 



DISTANCE OF A GIVEN POINT FROM A GIVEN LINE. 57 

We may now regard the equation of a line, when in the form 
x cos a -\-y sin a — p = 0, as expressing that the point whose co-ordi- 
nates are x and y : is at no distance from the line; that is, on the 
line; and when, on substituting the co-ordinates of any given point 
for x and y, the equation is not satisfied, the value of the first mem- 
ber expresses the distance of the point from the line, and its sign 
shows on which side of the line it is situated. 

75. In applying this formula, it is necessary first to reduce the 
given equation to the proper form, by dividing through by the 
square root of the sum of the squares of the coefficients, as ex- 
plained in Art. 52. But as we are not concerned with the value 
of a, it is not necessary to attend to the sign of p, as in Art. 54. 
Thus, given 12^ — 5x -j- 26 = 0, we divide by 13, and the result 
is i*y _ _^ x _l. 2 = 0, therefore i|/ — -fyx' + 2 =p' is the 
formula for the perpendicular from a given point. If we require 
the perpendicular from the point (3, 2), substitution of the values 
gives p' = 2 -^ ; for the point (1 — 2), p' = — -^. The opposite 
signs of these values show that the points are on opposite sides of 
the line. If we make x' = and y' = 0, we shall have the abso- 
lute term, which is therefore the perpendicular from the origin; in 
this case it is two units in length and has the positive sign. As 
all points on the same side of the line have perpendiculars of the 
same sign, we find that a point is on the same side as the origin, when 
the perpendicular has the sign of the absolute term; thus (3, 2) 
is on the same side of the line with the origin, but (1, — 2) is on 
the side remote from the origin. 

Examples. — Find the length of the perpendicular from the 
point (2, 3) on the line 2x -j- y — 4 = 0. 
3 

Ans. —7^, and the point is on the side remote from the origin. 

Find the perpendiculars from each vertex to the opposite side of 
the triangle, whose vertices are (1, 2),( — 2, 0) and (6, ■ — 1). 

Supposing the axes rectangular, what is the general expression 
for the perpendicular from P' on any line ? 

A*' -f- By -f c 



Ans. 



l/A 2 + B 2 
What is the distance of P' from the lines x = and y = ? 



C* 



58 THE STRAIGHT LINE. 

Formula for Line Bisecting the Angles of Given Lines. 

76. We have seen that Ax -f By -f C -f k ( A'x + B'y -f C) = 
represents a line, passing through the intersection of two given 
lines, k being an arbitrary constant, the value of which we deter- 
mined in the examples of Art. 60 by equations of condition. If the 
equations of the given lines be put in the form x cos a -\- y sin a — 
p = (the axes being still supposed rectangular), we have 

x cos a -\- y sin a — p -\- k (x cos a! -f- y sin a' — p r ) = 0. 

We have now proved that the expression x cos a -f- y sin a — p 
is the value of the perpendicular from any point upon the line, 
whose equation is x cos a -\- y sin a — p = ; that is, on the first 
of the two given lines. The expression in brackets is the perpen- 
dicular upon x cos a' -J- y sin a' — p' = 0, the other given line. 
Therefore the equation above asserts that the perpendicular from 
P, on the first line, equals — k times the perpendicular upon the 
second. Thus, IZx -f- by -f 9 = and 4x — 3y -f 10 = being 
the given lines, if we reduce the equations to the proper form and 
combine them, we shall have \^x -|- -^y -j- -fy -j- k (-Ja: — fy-\-2) 
= 0, expressing the condition that the perpendiculars on the two 
lines shall be in the constant ratio of — k : 1. 

It is easy to see, in general, 
that this condition is fulfilled by v \t' 

all points on a certain line, pass- \. r^. 

ing through the intersection of \ „ ^^^--^^\ 

the given lines. For let NR ^^^^^i \ p 

and NT be the given lines, and ^^^^ ^v 

NP any line passing through N, ^s/ 

then the ratio of the perpendicu- \ 

lars, PB and PT, from any point 

of this line is the same as the ratio of the sines of the angles PNB, 

PNT, into which the whole angle BNT is divided. 

77. The position of the line PN depends upon this ratio of per- 
pendiculars ; that is, on the value of k. If k = 1 or k = — 1, the 
perpendiculars are equal, and either the angle BNT or the supple- 
mentary angle BNT' is bisected ; therefore 

x cos a -j- y sin a — p ± (x cos a! -\- y sin a' — _p') = 



ETC. 59 

are the equations of the lines bisecting the angles of the given 
lines. The position of the lines with respect to the axes is imma- 
terial, except as the position of the origin determines the sign of 
the perpendicular. Thus, in the figure, suppose the origin to be 
situated within the angle RNT, and the absolute terms to have the 
same sign ; then the perpendiculars PR, and PT (which, by Art. 
75, will have the signs of the absolute terms) will have the same 
sign, and k must be negative. That is, the absolute terms in the 
combined equations having the same sign, h must be negative for a 
line passing through that angle in which the origin is situated. 
Thus, in if x -f p, + ^ + k (ix - \y + 2) = 0, negative 
values of k give lines passing in the neighborhood of the origin. 
This criterion is readily remembered by considering the value that 
h must have in order that the line pass through the origin — namely, 

Q 

k = -, as explained in Art. 62. 

Now giving k the values -J- 1 and — 1, in the example, and 
clearing of fractions, we have 112x — 14y -|- 175 = and 8x -J- 
64?/ — 85 == for the equations of the two bisectors of the angles 
of 12x -\- oy -\- 9 = and 4x — 3y -j- 10 = 0. Since these lines 
bisect supplementary angles, they are mutually perpendicular ; ac- 
cordingly, on reducing them to the form y = mx -\- b we find the 
value of m in the first to be 8, and in the other — i, which is its 
negative reciprocal, as in Art. 48. 

78. In general, the equations of any two lines referred to rec- 
tangular axes being Ax -f- By -j- C = and A'x -j- B'?/ -j- C = 0, 
reducing them to the proper form and combining, making k= -+-.1, 
and finally clearing of fractions, we have the equations 

l/A' 2 + B' 2 (Ax + By + Ol) ± l/A 2 + B 2 (A'x + B'y + C) = 

for the lines bisecting their angles. If now we take the two values 
of m from these equations and multiply them together, we shall find 
their product to be — 1, showing that they are of opposite signs, 
and reciprocal, which is a general proof that the lines are perpen- 
dicular.* 

* By trigonometric analysis it may be shown that x cos a -\- y sin a — p 
zfc (x cos a f -\- y sin a — p ; ) = have the inclinations 90° + \ (a -j- a r ) and 
\ (a -\- a'). — Chauvenet's Plane Trigonometry, Eqs. Ill and 114. 



60 THE STRAIGHT LINE. 

Examples. — Find the line bisecting that angle of 3x -f- 2y -j- 
5 = and x -j- y -f- 2 = 0, which contains the origin. 

Arts, (3l/2 — i/l3)x+(2i/2 — l/l3)^ + 5i/2 — 2l/l3 = 0. 

Find bisectors of the angles of 2x -j- y + 8 — and x -\- Zy — 
3 = 0; of the angles of 7x -j- y = 3 and x =y. 

Ans. \2x — % — 3 == and 2x -f 6y — 3 = 0. 

Find lines bisecting the angles which Sx — \y -\- 15 = makes 
with the axis of X (.y =--0). 

J.9W. x — 3y -f- 5 = and 3x -j- y -f 15 = 0. 

Find bisectors of the angles of these last lines, and finally bisectors 
of the results. 

Notice that the equations of perpendicular lines when in their 
simplest forms may be added and subtracted at once, the radicals 
which reduce the equations to the proper form being the same. 
The bisectors of the angles of perpendiculars are inclined to them 
at angles of 45°, and if we repeat the operation on them we return 
to the original lines. 

Prove this to be generally true by means of the equations y — 
mx — b = and my -j- x — a = 0. 

Equations Representing Two or more Lines. 

79. If from the equations of two loci, a third equation be derived 
by any algebraic process, its locus will pass through all the points 
of intersection of the original loci. For the original equations are 
both true of any one of these points, hence the derived equation 
must be true also. 

For example, y = 2x -J- 3 and y = x -\- 1 represent certain 
straight lines : multiply them member by member ; the result is 
y 2 = 2x 2 -j- 5x -f- 3, which represents a curve, of which all we 
know at present is, that it passes through the intersection of the 
given lines, which may be easily verified. 

We have already examined the effect of adding the equation of 
straight lines, and found a general equation including all the results 
which can be derived in that way, introducing an arbitrary multi- 
plier k, because the result is affected by previously multiplying one 
of the equations through by any number, or both of the equations 
by different numbers. 

80. If we are to multiply two equations member by member, it 



EQUATIONS REPRESENTING TWO OR MORE LINES. 61 

is plain that previously multiplying one of them by any number, 
has no effect upon the locus represented by the result • but previous 
transposition of a term from one member to the other has an effect 
upon the result. 

We now suppose all the terms transposed to one member in each 
case, and these members multiplied ; for example, the above straight 
lines would thus give (y — 2x — 3) (y — x — 1) = 0. Such an 
equation will be satisfied, not only by the point or points which 
satisfy both the given equations, but by all points which satisfy 
either, and by no other points. For to satisfy the equation, one or 
other of the factors must become zero ; that is, the point must be 
on one of the given lines. 

In general, if S = and S' = represent the given equations 
(S and S' representing polynomials of any degree containing x and 
y), SS' = is satisfied by all points, which make either S = 0, or 
S' == 0, that is, by all points of both the given loci. We shall 
therefore call it the compound equation of the given loci, reserving 
the term combined equation for S -j- &S' == 0, which is satisfied by 
those points which make S = and S' = simultaneously, and 
also by certain points which make neither S = nor S' = 0. 

81. The general compound equation of two straight lines is 

(Ax + By + C) (A'* + B'y + ,00 = : 

when expanded we find it to be of the second degree, thus the 
compound equation of y — 2x — 3 == and y — x — 1 = is 
y 2 — 3xy -j- 2x 2 — 4y -f- 5x -f- 3 = 0. We see therefore that an 
equation of second degree may represent a pair of straight lines. 
Given the above equation, we should know its locus completely, if 
we could resolve the first member into its factors ; but an expression 
of the second degree, containing x and y, is not generally resolvable 
into factors; the condition on which it is resolvable belongs to the 
discussion of the general equation of second degree. 

82. If the two factors are the same, that is, if the expression is a 
perfect square, the two lines coincide j thus 

(Ax + By + C) 2 = 

is satisfied only by the points of a single line ; but being an equa- 
tion of the second degree, its locus is said to be "a pair of coinci- 



62 THE STRAIGHT LINE. 

dent lines •" thus x 2 — 4xy -f- 4y 2 = is the equation of two lines 
coincident with x — 2y = or x = 2>y • x 2 = 0, of two lines coinci- 
dent with the axis of Y. 

In some instances the equation can be reduced by the ordinary 
methods of solution, or the expression in the first member can be 
factored at sight : thus x 2 =y 2 reduces to x = ±y, x 2 — 3x = — 2, 
to x=l or x = 2-, and x 2 — y 2 = is equivalent to (x-\-y) 
(x—y)=.0, x 2 — 3x + 2 = 0, to (x — l)(x — 2) = 0, xy — 
lOx = 0, to x (y — 10) = 0. 

Examples. — Form the compound equation of 2x — y = 3 and 
y — 4x = 6 ; of the parallel lines y = mx -j- b and y == mx — b ; 
of x = a and y = b; of the perpendiculars y — mx — b = and 
my -\- x — a = 0. 

What is the locus of xy = ? of x 2 — % 2 ? of y 2 = 3y ? 



CHAPTER III. 

TRANSFORMATION OF CO-ORDINATES. 

83. Having found the equation of a locus as referred to certain 
lines as axes, it may be desired to find its equation as referred to 
other axes, or to a new system of co-ordinates. In other words, 
given a relation between certain co-ordinates of a point, let it be 
required to find an equivalent relation between other co-ordinates 
of the same point. 

The co-ordinates which occur in the given equation are called the 
old co-ordinates, the others the new co-ordinates; if now we find 
values of the old co-ordinates in terms of the new, and substitute 
them in the given equation, the result will be an equation between 
the new co-ordinates, which will be true of every point for which 
the old equation was true. Hence it will be the required equation • 
in finding it we are said to transform the equation to new co-or- 
dinates. 

84. The old co-ordinates or variables of the given equation being 
denoted by x and y, we shall for distinction use X and Y to denote 
the new co-ordinates or variables which will appear in the new 
equation. Certain constants determining the position of the new 
axes relatively to the old ones must be introduced, which are called 
constants of transformation. To simplify the preparation of formulae, 
we consider first the case in which the origin is moved, afterward 
those in which we pass to the system of polar co-ordinates, or change 
the direction of the axes. 

85. CASE I. — To pass to a system of parallel axes 

WITH A NEW ORIGIN. 

In the figure let be the old, and 0' the new origin ; let x' 

and y' be the co-ordinates of 0', relative to the old axes. Take any 

point, P, and draw PR, or y, its ordinate, to the old axes ; it is 

divided by the new axis of X into two parts, of which one is the 

63 



64 



TRANSFORMATION OF CO-ORDINATES. 



new ordinate of P or Y, and the other equals the old ordinate of 
O'ory. Similarly, a; 'con- 
sists of two parts, X and / / p 
x' ' ; hence the formulae, 



O' 



For example, to trans- 
form an equation to paral- 

lei axes intersecting in 
the point (4, 3), we have 
a;=X-f 4andy=Y + 3 
for formulae of transfor- 
mation. Applying them 

to the equation of the straight line x -f- 2y = 10, gives X -j- 4 -j- 
2Y -f 6 = 10 or X -f 2Y == 0. This equation is true with re- 
spect to new co-ordinates, wherever x -J- 2y = 10 is true of the old 
co-ordinates. The want of an absolute term shows that the line 
passes through the new origin, as may easily be verified, and we 
observe it to have the same direction ratio, as we should expect 
since the axes are parallel. 

Examples. — Transform 2x -j- 3y = 6 to the new origin (1, — 2), 
etc., etc. 

Transform xy -J- 2x — y = 2 to the origin (1, — 2). 

Transform y — y' = m(x — x') to the new origin P' (x',y'). 

Transform x 2 -\- y 2 = 25 to the origin ( — 4, 3), and the result 
to a third origin whose co-ordinates referred to the second axes are 
(7,-7). 

Find the co-ordinates of this third origin as referred to the 
original axes, and verify by transforming the original equation 
directly. 

86. CASE II. — To pass from rectangular to polar co- 
ordinates. 

We here suppose the pole to coincide with the origin, and the 
initial line with the axis of X. Then drawing the ordinate and 
radius vector of any point, P, we have the right-angled triangle 
PRO, in which the angle POR is denoted by 6. From the defi- 



nitions of trigonometry, cos 6 = - and sin d 



- : hence 
r 7 



TRANSFORMATION OF CO-ORDINATES. 



65 



r cos 0, y = r sin 0. 





Y 








P 






r/ 








X \ 


y 











v 




x K 





Owing to the sim- 
plicity of these formulae 
the polar and rectangu- 
lar equations are usually 
treated in connection. 

In the figure r is posi- 
tive, because is always 
the angle between the 
positive directions of r 

and x; therefore the functions of are positive in the first quad- 
rant, and in all the quadrant follow the signs of x and y. If we 
consider r, in the figure, as negative, must be increased by 180°; 
cos and sin will then both be negative, and the formulas still 
give positive values for x and y. 

Examples. — Transform from rectangular to polar co-ordinates 
x 2 +y 2 = 3Q> 2x—y = 10;x=y, 

Ans. r = and indeterminate (the origin), or else = 45° 
(straight line through the origin) ; y = 2x, 

Ans, r sin = 2r cos or tan = 2, 6 = 63° 26' 6". 

87. If the pole is to be at any other point than the origin, we 
must add its co-ordinates, x' and i/, to the above values of x and y ; 
or, which is more simple in practice, transform by Case I. to the 
given point as origin, and then pass to polar co-ordinates. If the 
initial line does not coincide with the axis of X, let a denote its 
inclination, then the angle POE, will be -j- a instead of 6. Hence 
the formulas become 



x = r cos (0 -f- a), y = r sin (6 -j- a). 

For example, to transform to a system in which the initial line 
makes an angle of 45° with the axis of X, the formulae are x = 
r cos (0 -f 45°), y = r sin (0 -f- 45°). By the rules of " angular 

y f 

analysis" they may be expanded to x = (cos 6 — sin 0), y = 

l/2 l/2 

(cos 6 -f- sin 0). — [See Chauvenet's Plane Trig., Eq. 149.] Ap- 

v 

plying them to the line x -{-y = 4, we find 2 cos = 4, or 

1/2 
6* 



66 



TRANSFORMATION OF CO-ORDINATES. 



r cos 6 = 2j/2 for its equation in the required polar system. The 
line is therefore perpendicular to the new initial line, and at a dis- 
tance 2f/2 from the pole, by Art. 71. 

Examples. — Transform x == 7 -{- y and 2cc == y, to the above 
system. 

88. CASE III. — To pass from one rectangular system 

TO ANOTHER WITH THE SAME ORIGIN. 

As rectangular co-ordinates are most frequently employed in the 
applications of analysis, we 
separate this case from the 
general one of change in 
the direction of axes, and 
treat it in connection with' 
polar co-ordinates and the 
angular analysis. 

Let a denote the incli- 
nation of the new axis of 
X to the old, and r and 
6, the polar co-ordinates of P, referred to it as initial line. Then 
X and Y denoting the new co-ordinates, we have X = r cos 6 and 
Y == r sin ; but expanding the values in the last Article, x = r cos 
(6 -f- a) and y — r sin (B -j- a), we have 





p 

/ \y 

/ y \ 


----''"'' 


\ x 





x=r cos 6 cos a — r sin 6 sin a, 
y = r sin cos a -(- r cos <? sin a ; 

substituting X and Y for their values, we have the required 
formulae, 

x = X cos a — Y sin a, 

y == Y cos a -\- X sin a. 



The student familiar with trigonometry may recall these formulae 
at any time, by connecting them with the formulae of expansion, for 
sine and cosine of the sum of two angles; remembering that x cor- 
responds to cosine, being measured horizontally, and y to sine, being 
measured perpendicularly. 

89. By drawing, from the foot of the ordinate Y, lines perpen- 
dicular and parallel to the old axis of X, it is easy to see that we con- 
struct lines equivalent to the terms of which x is the difference, and 



TRANSFORMATION OF CO-ORDINATES. 67 

to those of which y is the sum, according to the formulae. We have 
thus a direct proof of the formulae, which is, in fact, precisely the 
same as that by which the formulae of trigonometry above referred 
to was originally proved.* 

If the new axis of X is on the other side of the old one, a must 
be considered negative or in the fourth quadrant; that is, cos a re- 
mains positive, but sin a is negative, so that x is the sum of two 
terms and y the difference of two terms, as in the trigonometric 
formulae for cosine and sine of the difference of two angles. If 
a = 90°, the formulae become x = — Y, y = X, showing that the 
abscissa has become an ordinate and the reverse, but the negative 
direction of the new axis of Y corresponds with the positive direc- 
tion of the old axis of X. Similar interpretations may be given to 
the cases a = 180°, a = 270°, a = 360°. 

Examples. — Transform 2x — y = 3 from rectangular axes to 
others making a = 45° ; to axes inclined at 60° (sin 60° = 2 j/3 
cos 60° = £). 

Transform 3x = 2 — 4y by turning the axes back 30°(a == — 30°), 
and the result by turning them forward 90°. 

Verify by turning the axes of the original equation forward 60°. 

Transform x 2 -\-y 2 — 4cX — 6y = to the origin (2, 3) and axes 
bisecting the angles of the given axes. 

Transform x 2 -f- y 2 = R 2 by turning the rectangular axes through 
the angle 6. 

Ans. X 2 -f- Y 2 = R 2 ; the equation is not altered by turning the 
axes through any angle which can be a property of no line except 
a circle with its centre at the origin 

Transform x cos a -\- y sin a =jp in the same manner. 

Ans. By angular analysis the equation reduces to x cos (a — 0) -j- 
y sin (a — 0) =p, which is still in the form of Art. 51. 

90. CASE IV. — To pass from any Cartesian system to 

ANY OTHER WITH SAME ORIGIN. 

This is the general case of change of direction ; it will be neces- 
sary to introduce two constants of transformation — namely, angles 
determining the new directions of the two axes. Let a denote the 
inclination of the new axis of X, and /5 that of Y, to the old axis 

* Compare with the above figure, Fig. 10, Chauvenet's Plane Trig. 



Q K 



68 TRANSFORMATION OF CO-ORDINATES. 

of X; and let w denote the angle between the old axes. Draw 
the ordinates of P, and also from 
P a perpendicular PR, to the old 
axis of X, and from the foot of the 
new ordinate, Y, a perpendicular and 
parallel. By the right-angled tri- 
angles PQR, MON and PMS, which 
contain angles equivalent to w, a 
and ft, PR =y sin w, SR = X sin a, 
PS = Y sin ft ; hence 

y sin co = X sin a -j- Y sin ft. 

If a perpendicular be drawn from P to the old axis of Y, and a 
figure constructed in a similar manner (drawing the abscissas of P), 
we may prove 

x sin co = X sin (to — a) -j- Y sin (<*> — ft), 

to — a and co — ft being the inclinations of X and Y to this axis, as 
a and ft were to the axis of X. 

The formulae will be most easily remembered, in the above forms 
which express that " either co-ordinate, into the sine of its inclina- 
tion to the axis it cuts, equals the sum of the new co-ordinates 
multiplied each by the sine of its inclination to the same line j" 
both members of this equality being the true distance of the point 
from the axis. But the formulae for substitution are 

x = x sin (co — a) ■ y sin (co — g) 
sin (o sin m 

^ sin a TT sin ft 

y = X - — -f Y . 

sin co sin to 

The angles a and ft are between the positive directions of the 
axes ; if either of the new axes is on the other side of the old axis 
of X, the corresponding angle is negative, and its sine, and one term 
in the formula for y is negative. If one of the new axes is on the 
other side of the old axis of Y, co — a or co — ft becomes negative. 
If we make co = 90°, so that the old axes are rectangular, and 
ft = a -\- 90°, so that the new are also rectangular, the formulae re- 
duce to those of Case III. In using; these formulae the values of the 



TRANSFORMATION OF A POINT. 69 

four fractions should be first computed, and the formulae simplified 
before substituting. Thus, to transform from axes making an angle 
of 60° to axes bisecting their angles: here 10 = 60°, a == 30°, 

j3 = 120°: hence x =— — Y and y = — 4- Y. 

l/3 l/3 

Examples. — Transform by these formulae x 2 -J- y 2 = 100 ; y = 

a; -f- 3 ; x -\- y = 6. 

Transform the same lines to axes making a = 45°, {3 = 90°, 
(«, — 60°). 



Transformation of a Point. 

91. The formulae we have given for transformation are values of 
the old co-ordinates in terms of the new, because their principal 
application is to the transformation of equations. If we have the 
co-ordinates of a point in one system, and require its co-ordinates 
in a new system, it would be more convenient to have formulae, 
giving the value of X and Y, in terms of x and y. Such formulae 
we may call reverse formulae of transformation. Thus, to pass from 
a rectangular system to another inclined to it by 60°, the formulae 
of Case III. become 

x = i X — il/3 . Y, y = \ Y + Ji/3 • X. 

Solving these equations for X and Y, we have 

X = lx+ il/S.y, Y = \y — \VZ-x. 

If then we wish to transform the point (4, 2), we find by substitu- 
tion its new co-ordinates X = 2 -f-l/3, Y= 1 — 2l/3. 

92. These reverse formulae might be found also by interchanging 
x and X, y and Y, in the original formulae, and making the proper 
changes in the constants. Thus in Case I. they are X = x — x f , 
Y =y — -y r , in which the constants are — x' and — y\ because 
these are the co-ordinates of the old origin referred to the new axes. 
In Case III. a becomes — a, because we reverse the direction in 
which we turn the axes. 

Examples. — Transform (3, 3) from rectangular axes to axes 
bisecting their angles; the point (1, — 2) to the same axes. 

Transform (1/3, 1) by turning the axes three times in succession 
through 120°. The final result should be (i/3, 1). 



70 TRANSFORMATION OF CO-ORDINATES. 

Verify the reverse formulae of Case III., by finding X and Y by 
elimination. 

93. The equations of a point are in reality the equations of two 
lines of the forms x = a, y = b, parallel to the old axes, and inter- 
secting in the point. Substituting for x and y their values by the 
formulae of transformation, we have the new equations of these two 
straight lines, from which by elimination we can find the new co- 
ordinates of the point of intersection. The process is the same as 
when we find X and Y in terms of x and y, by combining the 
formulae ; the known co-ordinates a and b merely taking the place 
of the general co-ordinates x andy. Thus, in the example of Art. 91, 
the formulae are x = |X — il/3 . Y,y = £ Y -f- £l/3 . X ; hence 

JX — Ji/3.Y = a and iY + il/3.X=5 

are the new equations of the lines which determine the point (a, b). 
Their intersection is the point X = \a -\- $1/3.6, Y= ib — 
il/3 . a. 

94. If we put a= and 5 = 0, we have the new equations of 
the old axes, which would lead us to expect that the formulae would 
all be, as we have found them, linear, or of the first degree with 
respect to X and Y ; also, that in those cases where the origin is 
not moved, there would be no absolute term. Thus in Case 
III., if a = 45, the formula for x, is x =£l/2 (X — Y) and 
$l/2 (X — : Y) == 0~, or X = Y, is the new equation of the old 
axis of Y, or line whose old equation was x = 0. 

Transformation of Formulae. 

95. The method of transformation may be applied to formulae for 
lines in particular positions, so as to produce more general formulae 
for the same lines. Suppose, for instance, we know the form of the 
equation of a certain line, when referred to axes passing through a 
particular point, and we require its equation when this point has 
any position P', the former lines of reference being parallel to the 
axes. Thus, the equation of a circle referred to any rectangular 
axes passing through the centre is x 2 -f- y 2 = R 2 , what is the equa- 
tion when the centre is situated at P' ? 

Since the co-ordinates of the centre, x r , y' ', are given, we may re- 
gard that point as the new origin of Case I. ; and the given equa- 



ARBITRARY TRANSFORMATION. 71 

tion must be written X 2 -f- Y 2 = R 2 , because it is a relation be- 
tween co-ordinates, as measured from that point. We thus have 
the new equation to transform back to the old axes by the reverse 
formulae, 

X = x — af, Y=y-y', 

which gives (x — x') 2 -\- (y — y'~) 2 = R 2 , for the circle whose centre 
is P'. Thus, having the equation of a line, a particular point con- 
nected with it being at the origin, we find a more general formula 
by substituting x — x' and y — 1/ for x and y ; the new constants 
x' and y being the co-ordinates of the particular point. 

We have already met with an instance of this in the equation of 
the straight line, which is y = mx, a point of the line being at the 
origin, and y — y' = m(x — x'), a point of the line being at P'. 

96. In the formulae derived on this principle, P' will not gener- 
ally be a point on the line ; but, as above, in the case of the circle, 
the point to which, as origin, it is most readily referred. We shall 
frequently use X and Y for co-ordinates measured from this point 
or central co-ordinates, regarding them as abridged symbols for the 
differences of co-ordinates of point and centre, x — x' andy — y', 
to which they are equivalent. See Art. 101. 

97. The formulas of Cases III. and IV. might be used in a 
similar manner when the direction of the axes is changed, but not 
so conveniently, because the constants introduced would be angles. 
By Case II. general polar equations may be found from the general 
rectangular equations 3 thus, Ax -f- By -f- C = gives r (A cos 6 -j- 
B sin d) -\- C = 0. Also, when a polar equation can be put into 
a form in which r cos 6 and r sin 6 occur, it can be changed to a 
rectangular equation ; thus, r cos (0 — a) = p expanded is r cos d 
cos a -j- r sin 6 sin a =p, or x cos a -\- y sin a =p. In equa- 
tions of the second degree an additional formula for this reverse 
transformation, 

r 2 = x 2 +y 2 , 

will be useful. Thus, r = 5, or r 2 = 25, becomes x 2 -j- y 2 = 25. 

Arbitrary Transformation. 

98. The equation of a line or curve depends partly upon its 
shape, and partly upon its position with respect to the axes. The 



72 TRANSFORMATION OF CO-ORDINATES. 

circumferences of circles with different radii, for instance, are of 
different shapes, and hence their equations differ essentially; 
but the equations of the same circle in different positions are such 
that one may be derived from another by transformation. So that 
given the equation of a line, it is possible that with other axes it 
might have a simpler equation, from which its shape might be more 
easily deduced. 

To find these simpler equations, we use the formulae of transfor- 
mation in their algebraic forms, considering the constants as arbi- 
trary, and then examine the transformed equation, to see in what 
manner it can be simplified by particular determinations of the 
constants. 

99. This general transformation is of two kinds : first, change of 
origin. For example, 2x — 3y — 4 = 0, by substituting the formulae 
of Case I., becomes 2X — 3Y -j- 2xf — Sy' — 4 = 0, in which we 
can make the absolute term disappear by giving x' and y' such values 
that 2x' — 3y f — 4 = 0. Thus, by making the new origin satisfy the 
original equation, the equation of the line takes the simpler form 
2X — 3Y=0, which shows that the line passes through the new origin. 
The second kind is change in direction of axes, by the formulae of Case 

IV. For example, by this transformation y = 3x becomes XI 

\sin a) 

sin (w — a)\ /sin /? sin (a> — /9)\ „ 

3 ^ ) -f- Y ( - — — 3 — \ = 0. If we give a 

sin to I \ sin w sin w I 

such a value as to make the coefficient of X equal zero, the equation 

will reduce to Y = ; that is, the line will coincide with the new 

axis of X. The condition is = 3, which is satisfied 

sin (to — a) 

when a equals the angle of the line's inclination. (See note to Art. 48.) 
Examples. — To what origin must we transform x 2 -\-y' 1 — 

4x -f- 2y — 4 = to make it take the form X 2 -f Y 2 = R 2 ? to 

what, in order to make it have no absolute term ? 

The axes being rectangular, through what angle must we turn 

them to make 4x =y -f- 1 take the form Y = b ? (by Case III.) • 

100. This method may be used to simplify general equations. 
The general equation of first degree, Ax -f- By -\- C == 0, may by 
change of origin be freed from its absolute term, and then, by change 
of direction of axes, be reduced to the form Y = 0, which repre- 



ARBITRARY TRANSFORMATION. 73 

sents the axis of X ; that is, the axis can always be made to coin- 
cide with Ax -j- ~By -f- C — 0. The lines represented by this 
equation differ only in position, being in fact all straight lines ; but 
the lines represented by general equations of higher degrees differ 
not merely in position, but essentially in shape. 

The constants of transformation are four in number, x! and y f , 
which are introduced by the first transformation, and the angles a 
and /? by the other (w is not arbitrary, as it depends on the old 
axes). These constants are to be determined by the conditions 
that the coefficients of certain terms in the transformed equation 
shall be zero, so that those terms may drop out of the equation. 
The degree of the equation, however, cannot be altered; for, 
the values of x and y in the formulae being of the first degree 
with respect to X and Y, transformation cannot introduce terms of 
a higher degree, or raise the degree ; neither can it lower the de- 
gree, because transformation back, which should restore the original 
equation, cannot raise the degree. Hence the degree of the equa- 
tion is fixed, and naturally becomes a basis for the classification of 
lines. The formulae of Case II. show that the degree of a polar 
equation with respect to r is the same as that of the rectangular, 
and hence of any Cartesian equation of the same line. 

It may be shown that the absolute term, for any origin, P', is the 
result of substituting x' and y' for x and y, in the first member of 
an equation, whose second member is zero. For suppose Ax m to be 
a term ; in the transformed equation, we have A(X -|- x') m , expand- 
ing this by the binomial theorem, the last term Ax' m is part of the 
new absolute term. In like manner, each of the transformed terms 
contains an absolute part ; and the new absolute term consists of all 
these parts, together with the old absolute term. Compare the ex- 
ample in the last Article. 

7 D 



CHAPTER IV. 



THE CIRCLE. 

101. In treating the circle we shall use rectangular axes only, 
because the rectangular equation of this curve is the most simple 
and best adapted to investigating its properties. Its oblique equa- 
tion will afterward be given, in connection with the general equa- 
tion of a class of curves of which the circle will be found to be a 
special case. 

If the centre of the circle be at the origin, the equation is 
x * ~h y % = R 2 5 where R is the radius, and it was shown, in Art. 95, 
that consequently the general equation, when x' and t/ denote the 
co-ordinates of the centre, is 



(x 



7'+(y— •)•=* 



To make the meaning of 
the equation perfectly clear, 
draw a circle from any point 
P' as centre, and with a radius 
whose length is denoted by R. 
Let the co-ordinates of any 
point of the circumference, as 
P, be x and y\ and let the 
lines P'R and PR, which are 
the co-ordinates of P as mea- 
sured from the centre, or cen- 
tral co-ordinates of P, be denoted by X and Y 
right-angled triangles PRP, PR 2 -f- PR 2 = P'P 2 

X 2 -f Y 2 =R 2 




Then by the 
hence 



is the central equation; and since, in general co-ordinates, P'R is 



denoted by the difference x 

74 



x f , and PR by y — y' , the above is 



THE CIRCLE. 75 

the general equation. If P be taken in a different part of the 
circumference, the differences x — x' and y — y' may become nega- 
tive, but their squares will still be positive and equivalent to 
P'R 2 and PR 2 ) hence the equation is true for all points of the cir- 
cumference. The centre P' may be so situated that one of its co- 
ordinates is negative, in which case x — x' or y — y' will become 
the sum of the variable and constant, as in the equation of the 
circle whose centre is ( — 2,1), and whose radius is 3; namely, 
(x + 2) 2 + (y-l) 2 = 9. 

Examples. — Form the equation of the circle whose centre is 
the point (4, — 3), and radius 5 ; centre at (3, 0), and radius 2; 
etc., etc. 

102. In the equation of a circle, as indeed in all equations of 
the second degree, each value of x has corresponding to it two 
values of y, since substituting a given value for x gives a quadratic 
equation to find the value of y. For instance, in the circle 
(a- _ 4)2 _|_ r y _ 2)2 = 9^ i e t x== 2, the result is (y — 2) 2 == 5, 
hence y = 2 H- y 5. Either of these values of y, in connection 
with x = 2, will satisfy the equation, as is easily verified. We 
thus find two points of the line having the same abscissa ; x = 3, 
x = 4, etc., give likewise two values of y. But x = 7 gives 
(y — 2) 2 = or y = 2, a single value, and x = 8, gives y = 2 ± 
V — 7, which are imaginary values of y ; hence there is but one 
point having the abscissa 7, and no points having the abscissa 8. 

103. In general, solving the equation for y, we have 



y=y' ±l VR 2 — (x — a/) 2 , 

which is the equation of the circle, expressing y in terms of x, or 
making it an explicit function. On account of the double sign, we 
say there are two values ofy corresponding to each value of x) but 
when (x — x'y is greater than R 2 , these values are imaginary, and 
when (x — a/) 2 = R 2 , they are equal. Imaginary values will take 
place when the quantity under the radical is negative, or x — x' is 
numerically greater than R ; and equal values, when x = x' ± R. 
These latter values are, therefore limiting values of the abscissa of 
P, corresponding in the figure to the points A and B. Thus, in 
the example of the last Article, the limiting values of x are one, and 
seven units. There are no points of the curve, on the left of the 



76 THE CIRCLE. 

point (1, 2) or on the right of (7, 2), but if the abscissa be assumed 
between these values, two points of the curve will be found. 

In like manner, two values of x correspond to an assumed value 
of y, and the limiting values of y are those which make the. radical 
part of the value of x equal zero — namely, y = y' ± R ; these 
values belonging to the highest and lowest points, for each of which 
x = x'. We find, therefore, that an equation of the second degree 
may represent a line limited in all directions, unlike the equation 
of first degree, which represents an unlimited line. 

104. If the centre is at the origin, y = ±1/ R 2 — x 2 , or the two 
values of y, corresponding to a given value of x, are equal with con- 
trary signs. The two points having the same abscissa are situated 
at equal distances above and below the axis of X ; the curve is 
therefore said to be symmetrical to this axis, x 2 -\~y 2 = W is also 
symmetrical to the axis of Y. 

If the centre is on the axis of X, and at a distance from the 
origin equal the radius, the equation becomes (x — R) 2 -\- y 2 = R 2 , 
which is readily shown to be symmetrical to the axis of X, but not 
to that of Y. In fact any equation containing y 2 , and not contain- 
ing y in the first power, represents a curve symmetrical to the axis 
of X, for it is satisfied by equal positive and negative values of y. 

Various Equations of the Circle. 

105. The formula, which we have found for the circle, contains as 
constants, the radius and co-ordinates of the centre, and enables us 
to form the equation when these constants are given. If the centre 
and a point of the circumference are given, x' and y' are known, and 
R 2 may be determined by an equation of condition. For example, 
if the centre is to be the point (3, — 1) the formula gives 
(a; — 3) 2 -f ( y -f l) 2 = R 2 ; and if (2, 1) is to be a point of the 
curve, the equation of this condition is found by substituting these 
last values for x and y, which gives 5 = R 2 , hence (se — 3) 2 -J- 
(y -\- l) 2 = 5 is the required equation. In general, P' being the 
centre and P" the point of the curve, the equation of condition is 

(x "_., T + (/ _ /) , = I . ! . 

This is the same thing as finding R, or the distance between P" 
and P', by the formula of Art. 12; and the equation of the circle 



VARIOUS EQUATIONS OF THE CIRCLE. 77 

itself may be regarded as expressing that the distance between P 
and the fixed point P', shall be constant and equal to R. 

Examples. — Give the equations of the circle having (1, 2) as 
centre and (3, — 2) in the circumference ; (3, — 2) as centre and 
(1, 2) in circumference; ( — 2, 5) as centre and passing through 
(1, 1); etc., etc. 

106. When the properties of the curve are investigated, the 
algebraic work is simplified by using the central equation X 2 -f- 
Y 2 = R 2 ; while at the same time the results are applicable to any 
circle if we consider X and Y as standing throughout for x — x' 
and y — y' '. Thus putting the central equation in the form 

• Y 2 = R 2 — X 2 = (R -f X) (R — X), 

and observing that R -f- X represents the line AR, and R — X the 
line BR in the figure, we have the property of any circle, that " the 
square of the perpendicular, dropped from any point of the circum- 
ference on a diameter equals the product of the segments into which 
it divides the diameter." 

107. Expanding the equation in terms of centre and radius, 
we have 

x * + y 2 — 2x'rz — 2y'y -f- x n -f y' 2 — R 2 = 0, 

an equation containing x l and y 2 followed by terms of the first de- 
gree, and a constant part which may be represented by a single 
letter, or absolute term. Any equation of the form, 



Ax 2 -f Ay 2 -f T>x -f Ey -J- F = 0* 
may be reduced, by dividing by A, to the form, 

x 2 +y 2 + dx + ey+f=0, 

in which d, e and/* stand for the ratios — , etc. The first of these 

A 
is the general equation of the circle, in which the position of the 
line depends upon the ratios of the constants ; the second a form in 
which it depends upon the values of the constants. To construct a 
circle from its equation in this form, we must find the centre and 

* This form has been chosen, so that the letters shall stand in the same 
places as in the general equation of the second degree. 



78 THE CIRCLE. 

radius. Comparing it with the expanded form above, we see that 
the coefficient d takes the place of — 2x', e = — 2if and/= x n -\- 
y' 2 — R 2 , hence 

x' = - id, y' = - \e, R 2 = x' 2 -f f -f. 

Thus, given the equation 2x 2 -|- 2y 2 — 4x -|- Yly — 32 = 0, which 
is in the general form, divide through by 2, and we have x 2 -\-y 2 — 
2x -f 6y — 16 = 0, in which d = - - 2, e = 6 and /== — 16. 
Hence, x' = 1, y ! = — 3 and H 2 == 26 ; and tho equation can be 
written (x — l) 2 -j- (jf -f- 3) 2 = 26, the centre being the point 
(1, — 3) and the radius 1/26. Whatever the values of x' and //, 
the centre is readily constructed ; but if the value of R 2 , as found, is 
negative, R is imaginary, and no circle can be constructed. 

Examples. — Reduce 3x 2 -j- 3y 2 -f- 9x — 2y = 0, and — x 2 — 
y 2 -J- 2x-\- 4y — 10 = 0, to the first form of the equation of circle 
and construct if possible. 

108. We see then, that the general equation of the circle in- 
cludes certain equations, which cannot be constructed in this man- 
ner, because the expression for the radius is imaginary. These 
equations are said to be the equations of imaginary circles, and are 
not satisfied by any points. For such an equation may be reduced to 
the form (x — x f ) 2 -j- (y — y f ) 2 = a negative quantity, which can be 
satisfied by no values of x and y, because each of the squares in 
the first member is essentially positive, and their sum cannot be 
negative. If the equation takes the form 

(*-*T + (y-/) 2 = o, 

it is satisfied only by x = x f ,y=y', which makes each of the 
squares zero. We have here an equation satisfied only by a single 
point, but being of the form of the circle, and the radius taking the 
value zero, it is said to be the equation of an infinitesimal circle. 

109. Suppose now, the absolute term f to vary, the other con- 
stants in the equation remaining fixed. Since x' and y' depend 
only upon d and e, the centre of the circle remains in the same 
position, the radius only changing. We have, therefore, the equa- 
tions of a series of concentric circles. Now the radius (putting — \d 
and — \e in place of x and y), is 

R = V\(d 2 + c>)-f. 



VARIOUS EQUATIONS OF THE CIRCLE. 79 

If f is negative or zero, R, is real, since \ (d 2 -f- e 2 ) is essentially 
positive. But if / becomes positive and increases, R decreases, 
until /= | (d 2 -j- e 2 ), when R = 0_; and if /increases beyond this 
value, R becomes imaginary. 

110. If we wish to find the equation of the circle passing through 
given points, we must assume the form 

a? + tf+dx + ey+f=0 : 

in which the values of the constants determine the line. As there 
are three arbitrary constants, three equations of condition may be 
satisfied, or the circle may be made to pass through three given 
points. The number of arbitrary constants is always the same as 
the number of points through which the curve may be made to 
pass, but the general equation contains always one more constant. 
See Art. 56. 

For instance, let a circle be required to pass ^through (2, 1) 
( — 1, 3) and (1, — 1), the equations of condition are 



5 + 


2d + 


«+/= 


= o, 


10 — 


d + 


3*+/= 


= 0, 


2 + 


d — 


«+/= 


,0. 



and 

Eliminating f, by subtracting the second and third from the first, 
we have 

_ 5 _|_ 3tf _ 2 e = 0, 

and 3+ d + 2e = 0, 

from which d = f, e = — If : and substituting in one of the first 
set of equations, f= — 4}. With these values, the assumed equa- 
tion becomes 

x 2 ■+ y 2 -f \x — lfy — 4£ = 0, or 

4^2 + 4j/2 _|_ 2x _ 7 y = 17 ? 

which may be verified for each of the given points. 

111. As the co-ordinates of the centre depend upon d and e, each 
of the two equations between d and e is equivalent to a relation 
between the co-ordinates of that point. Put — 2x for d and — 2y 
for e, and we have 

_ 5 _ Q X 4_ Ay — 0, 

and 3 — 2x — % = 0, 



80 THE CIRCLE. 

in which x and y refer to the required centre. The first (being 
derived from the first and second equations of condition) is the 
condition imposed on the centre, or the locus of the centre of a 
circle passing through the first two points; namely. (2, 1) and 
( — 1, 3). In like manner, the other is the locus of the centre of 
the circle passing through the first and third points (2, 1) and 
(1, — 1). The point in which they meet ( — J-, f) is the required 
centre. 

As the equations are of the first degree, the loci are straight lines, 
and they may be shown to be perpendiculars passing through the 
middle points of the lines joining the given points. (Compare Art. 
15.) If the lines were parallel, which would be the case if the 
three points were in one straight line, the centre could not be 
found ; that is, a circle cannot be made to pass through three points 
in one straight line. 

Examples— Find the circle passing through (3, 3) (2, — 1) 
and (0, 2); through (1, 1) (0, 2) and the origin. 

Find the locus of the centre of a circle passing through (1, 6) 
and (— 3, 2). 

Show generally that the locus of the centre, for two given points, 
bisects perpendicularly the line joining the points. (P x and P 2 
being the given points, show the equation of this perpendicular to 
be equivalent to the equation found as above from general equa- 
tions of conditions.) 

Polar Equations of the Circle. 

112. Using the formulae of transformation x = r cos 0, y = r 
sin #, in x 2 -f- y 2 — R 2 , we have r 2 = R 2 or r = ± R. This is 
the polar equation when the centre is at the pole. It expresses 
simply that the radius vector is constant and may be considered 
either positive or negative ; thus r = 5 and r = — 5 denote the 
same circle. By the same transformation, 

r 2 j^ (J cos _|_ G s i n (9) r _j_ f = o 

is the polar equation of a circle in any position. 

This equation being of the second degree, gives two values of r 
for each value of 0. It is shown in algebra, of quadratic equations 



POLAE EQUATIONS OF THE CIRCLE. 



81 




in this form (the second member being zero, and the coefficient of 
the square being unity), that the 
absolute term is the product of the 
roots. Therefore the product of 
the distances from any point, 0, to 
the points P, P, in which a straight 
line passing through cuts a cir- 
cle, is constant and equal to the 
absolute term of the equation, that 
point being the origin or pole. If, as in the figure, is outside 
of the circle, the two values of r, being a secant and its external 
segment, are measured in the same direction ; therefore their pro- 
duct is positive, or / is positive. If the pole is on the curve, one of 
the values of r, and consequently their product is zero, and f=0. 
If the pole is within the circle, the two values of r, being the segments 
of a chord, are measured in different directions, and/" is negative. 
When the pole is at the centre, d = and e = 0, the positive and 
negative values of r become equal to radius, and/= — R 2 ; hence 
r = ± R. 

113. A more useful form 
of the polar equation may 
be found, thus : Let r' re- 
present the distance of the 
centre from the pole ; and 
let the initial line pass 
through the centre. The 
rectangular equation is 

{x — iy + f = n\ 

the centre being at the point (/, 0). Expanding and transform- 
ing, we have 

r 2 — 2rr' cos + r' 2 = R 2 . 




The three lines r, r' and R form the sides of a triangle of which 
6 is an angle. Hence the equation expresses that " the square of 
one side of a triangle equals the sum of the squares of the other two 
sides, minus twice their product, into the cosine of the included 
angle." 

D* 



82 THE CIRCLE. 

If the centre is not on the initial line, let 6' denote the inclina- 
tion of r', so that / and 6' are the polar co-ordinates of the centre. 
Then the angle POP' is denoted b y 6 — d' ', and 



— 2rr f cos (0 — 6') -f r' 2 = R 2 



is the polar equation of any circle, in terms of the co-ordinates of its 
centre and its radius. 

Examples. — Give the polar equation of the circle whose centre 
is (45°, 7) and radius 10; centre (90°, 5) and radius 2; etc., etc. 

114. To express the value of r directly in terms of 0, we complete 
the square in the first member, by subtracting r n sin 2 6 (using the 
equation where 6 f — 0). Thus 

r 2 _ 2r/ cos 6 -j- r n cos 2 6 = R 2 — r' 2 sin 2 6 
r = r' cos 6 ± l^R 2 — r' 2 sin 2 0. 

If, as in the figure, r' ^> R, these values of r will become imaginary 
for some values of 6. For, if R 2 = r' 2 sin 2 6 ; that is, if sin = ± — , 

the radical part vanishes, and both values of r equal / cos 6. These 
values of are only possible when / ^> R, for the sine of an angle 
is always a proper fraction. When they are possible, they are 
limiting values of 0, one in the first and one in the fourth quadrant, 
and the equal values of?* which correspond to them are tangents to 

the circle. For if sin 6 ^> — , the values of r are imaginary. 
/ 

To find the length of the tangent from 0, or the equal values of 
r, we must find what / cos 6 becomes, for these values of 0. Kow 
if r' 2 sin 2 = R 2 , r n cos 2 =r n — R 2 , hence the length of the tan- 
gent is l/V 2 — R 2 . From this value, or from the value of sin 6, it 
is easily shown that a tangent is perpendicular to the radius at its 
point of contact. 

Again, multiplying the two values of r for amy value of 6, we 
have r n cos 2 6 — (R 2 — r n sin 2 6) = r n — R 2 . Or the product of 
the two values of r, is constant, as shown in Art. 112, and is equal 
to the square of the tangent. This product is the absolute term of 
the equation ; /, in the previous form, r n — R 2 , in the present, or 
x n — y' 2 — R 2 , in the first rectangular form. When the origin or 



POLAR EQUATIONS OF THE CIRCLE. 83 

pole is within the circle, the absolute term is negative and the tan- 
gent is impossible. 

Examples. — Reduce r 2 — 8r cos = 9 to the form r 2 — 2>V cos 
6 -j- r n = R 2 , and determine the length of the tangent from the 
pole, and the limiting values of 0. 

In —x 2 — f + Zx + fy — 10 = 0, 4.r 2 + 4y 2 +2x — 7y=17, 
etc., is the origin within or without the circle ? and if without, what 
is the tangent? 

115. If ;■' = R, the polar equation becomes 
r 2 — 2>R cos = 0, 
which is satisfied by r = 0, whatever the value of ; because the 
pole is on the circumference, therefore one of the values of r 
vanishes or becomes zero. Divid- 
ing by r, we have 

r = 2R cos 0, 

which gives the other value of r, 

and hence is the polar equation 

of a circle referred to a point on 

the circumference and a diameter. 

Or the two values of r may be 

found by making the supposition / = R in the two values of r t 

Art. 114, which gives r = R cos ± R cos 0. 

It may be remarked in general of polar equations of the second 
degree, that if the pole be placed on the line, the absolute term dis- 
appears and the equation may be divided through by r. The result 
is an equation of the first degree for r ; but we must remember that 
in dividing though by r we neglect one root of the quadratic equa- 
tion — namely, r = 0. 

If. in r = 2R cos : = 0, r = 2R, the diameter OA, measured 
along the initial line. This is the greatest possible value of r, and 
since any other value of r. as OP in the figure, equals this diameter 
into the cosine of the included angle 0, the triangle OPA is right- 
angled at P • or " the angle inscribed in a semi-circle is a right angle." 
If = 90°, this value of r is also zero, or the line perpendicular to 
OA at O is a tangent. If is in the second or third quadrant, r 
is negative, or the line must be produced backward through the 
pole to meet the curve. 




84 THE CIRCLE. 

Intersection of Circle and Straight Line. 

116. We saw, in Art. 39, that the intersection of the loci of 
equations of the first and second degrees may be in two points. 
The example in that Article was the intersection of the circle 
x 2 -|- y 2 = 25, with the straight line 25 -j- x == 7y, in the two 
points (3, 4) and ( — 4, 3). In Art. 40 the same circle is shown 
not to intersect with the line x = y — 8, by the occurrence of 
imaginary quantities in the simultaneous solution of the equations. 
A circle will therefore generally meet a line in two points, if it 
meet it at all, and it cannot meet it in more than two points. The 
same remark would apply to the intersection of the locus of any 
equation of the second degree, by a straight line. 

Tangency to a curve of second degree is indicated by the occur- 
rence of equal roots; because then, the line meets the curve in but 
one point. It is therefore a special case, interposed between the 
more general cases of intersection in two points, and non-intersec- 
tion. This is exemplified in Art. 102, where we found values of y 
corresponding to given values of x in the circle (x — 4) 2 -|- 
(3/ — 2) 2 = 9, which is the same thing as finding the intersections 
of the circle with the lines x = 2, ,x '== 3, etc. x = 7 was there 
the tangent ; x = 6 would give points of intersection, and x = 8 
imaginary values. 

117. In combining equations of the first and second degree, we 
should substitute the value of one variable from the first into the 
second, which gives a quadratic equation. Solving this, we obtain 
"two values of one co-ordinate, each of which must be substituted in 
the equation of the first degree, to find the corresponding value of 
the other co-ordinate. The work should then be verified by sub- 
stituting in the equation of the second degree, to see if the points 
found satisfy that equation. Thus, given the circle x 2 -f- y 2 — 
2x -j- y -j- 1 == 0, and the straight line x — y — 1. Substituting 
for x its value y -f- 1^ we have the quadratic 2y' 2 -f- ^ — Q. This is 
satisfied by y = and y = — J. The corresponding values of x, 
from x =y -f- 1, are 1 and J. Therefore the points are (1, 0) and 
(i, — £), which will both be found to satisfy the equation of the 
circle. 

As there are two solutions, which may be verified even when the 



INTERSECTION OF CIRCLE AND STRAIGHT LINE. 85 

values are imaginary, a line is said to meet a circle in two real, or 
coincident, or imaginary points, according as it is a secant, or tan- 
gent, or fails to meet the curve. 

Examples. — Find the intersections of x 2 -\- y 2 -j- 6x — 3y == 10, 
with the line x -j- y = 2 ; with the line y = 3x — 4 ; with 
x = 4; etc., etc. 

118. In finding general values of the co-ordinates of intersection, 
we shall for convenience use the central equation, 

X 2 -f Y 2 = R 2 , 

in which X and Y stand for x — x' and y — y'. For the straight 
line we shall use the rectangular form, 

X cos a -j- Y sin a = p, 

in which, as X and Y are measured from the centre, p is the per- 
pendicular from the centre. To eliminate Y between these equa- 
tions, multiply the first by sin 2 a, and substitute for Y 2 sin 2 a its 
value from the second : the result is 



X 2 sin 2 a -f (p — X cos a) 2 = R 2 sin 2 a, 
reducing to X 2 — 2pX cos a = U 2 sin 2 a — p 2 . 
Completing square by adding p 2 cos 2 a, we have 

(X — p cos a) 2 = (R 2 — p 2 ) sin 2 a, 
since p 2 =p 2 sin 2 a -f p 2 cos 2 a. Hence the values of X, 
X = p cos a zh l/(R 2 — p 2 ) sin a. 

Substituting these values of X, in Y sin a =p — X cos a, we 
deduce 

Y — p sin a =p l/(R 2 — p 2 ) cos a, 

for the corresponding values of Y, the upper signs being taken 
together for one point, and the lower signs for the other. The re- 
sult is verified by squaring the associated values of X and Y 7 , and 
adding; the sum is found in each case to be R 2 . 

119. These are the general values for the central co-ordinates 
of the points P, P, in which the straight line cuts the circle. It is' 
easy to see that the rational parts, p cos a and p sin a must be the 



86 



THE CIRCLE. 



co-ordinates of the middle point of the chord PP. But p cos a and 
p sin a are the co-ordinates of M, the foot of the perpendicular, p, 
A perpendicular from the centre therefore bisects a chord. In 




general, the rational part of the two values of each co-ordinate of 
intersection, being an arithmetical mean between them, is a co- 
ordinate of the middle point of the chord. See Art. 11. 

The real or imaginary character of the radical part shows 
whether the line cuts the curve or not. The supposition which 
makes the radical part zero is intermediate between those which 
make it positive and negative, and answers to the case in which 
P, P and M coincide, and the line touching the curve in a single 
point is called a tangent. This method of finding a condition of 
tangency — namely, to place the radical part equal zero — we shall 
apply to all curves whose equations are of second degree. In 
this case the condition is that p 2 — R 2 ; a less value of_p 2 will make 
the points P, P, real ; a greater value will make them imaginary. 

The condition of tangency will always be a relation between the 
constants occurring in the equations of the straight line and curve. 
Its simplicity will of course depend upon the form of the equations 
employed. We used the central equation for the- curve, so as to 
avoid introducing more than one constant relating to the circle. It 
happens that the constant a, from the line, does not appear in the 
condition, so that whatever the direction of the line it will touch 
the circle, provided only, its distance from the centre has the 
proper value. 



TANGENT TO THE CIRCLE. 87 



Tangent to the Circle. 

120. The condition of tangency, for the circle, being p 2 = R 2 , 
or p == zh R, substituting this value of p in the equation of the 
line gives the equation of a tangent, 

X cos a -j- Y sin a = zh R. 

This line will touch the curve, whatever the value of a, and the 
double value of p indicates that parallel tangents may be drawn 
through the two extremities of a diameter, DP X . The form in 
which p = -f- R, which, with the value of a in the figure, repre- 
sents the tangent at P 1} need only be considered, when all values of 
a from 0° to 360° are admitted ; for the tangent at D may be con- 
sidered as having a value of a, 180° greater than that at V x . 

Putting p = R, in the values of X and Y, we find the co-ordi- 
nates of P 1} 

X x = R cos a, Y x = R sin a. 

Finally, substituting x — x' for X, etc., in the equations of circle 
and tangent, we have, in general, co-ordinates, for the circle, 

the general equation of a tangent, 

■ (x — J) cos a -f- {jj — y ) sin a = R. 

This equation contains one arbitrary constant, a, to which any value 
may be given. Whatever this value may be, the line fulfils the 
condition of tangency to the given circle. Thus, given the circle 
x 2 -f- # 2 -j- 2:c — 4i/ — 8 = 0. Reducing the equation to the form 
containing x\y' and R, it becomes (x -f- l) 2 -\- (y — 2) 2 == 13. 
Making the substitutions in the general equation of tangent, we 
have (x -f- 1) cos a -{- (y — 2) sin a = 1/13, for the equation of 
any tangent to the given circle. 

Examples. — Find the general equation of the tangent to the 
circle x 2 -\- y 2 — 6x -\- 8y = 0. 

Find the equation of a tangent to this circle, parallel to 3y — 
4:X= 10, by giving cos a and sin a the same values as in this line. 



88 THE CIRCLE. 

Verify that the line found is a tangent, by finding its intersec- 
tions with the given circle. 

121. To find the equation of a tangent line, in terms of the co- 
ordinates of the point of contact, we may eliminate a, introducing 
X x and Y 1 , in the equation of tangent. 

From the values of X x and Y l5 we find 

X x . Y, 

cos a = — sin a = — , 

R' R' 

substituting which the equation of tangent becomes 

XX, + YY! = R 2 . 

This is the equation of a line touching the circle at the point 
P l5 which is supposed to be a point of the curve. Thus, (3, 1) 
is a point of the circle x 2 -J- y 2 = 10, for it satisfies its equa- 
tion. Hence 3x -J- y = 10 is the tangent to the circle at that 
point. It is easy to verify, both that this line passes through the 
given point and that it touches the circle. 

Examples. — Find tangents to x 2 -J- y 2 = 25, at the points of 
the curve whose abscissa is 4. 

Ans. The corresponding ordinate is zh 3, and the tangents are 
4x + 3y = 25 and 4x — 3y = 25. 

Find lines touching x 2 -\- y 2 = 9, at the points where it cuts the 
axis of X. 

It must be remembered, however, that the form of the above 
equation does not make P x a point of the line. X x and Y x merely 
occur in the equation as constants, and if they are substituted for 
the variables X and Y, in order to see whether P x satisfies the equa- 
tion, the result is 



Xi 2 + Y : 2 = R 2 . 

This is an equation of condition, true only when P x is a point of 
the circle. Therefore, XX X — YY X = R 2 is not the equation of a 
tangent unless P x is on the curve. We shall hereafter see what it 
represents in case P x is not a point of the curve. 

122. Since a straight line may be made to fulfil two conditions, 
a tangent line, which already fulfils one condition, may be made to 



TANGENT TO THE CIRCLE. 89 

fulfil another; for instance, that of passing through a given point. 
Suppose the equation of the circle to be given in its central form, 
then we may assume for the tangent the form X cos a -j- Y sin a = R, 
and determine a by an equation of condition. We need not deter- 
mine the angular value of «, of course, but only determine the co- 
efficients, cos a and sin a, by the equation of condition, and the rela- 
tion sin 2 a -f- cos 2 a = l. 

We may instead of this assume the form XX X -\- YY X = R 2 , and 
determine the constants X x and Y x by the equation of condition 
expressing that the given point is on the tangent, and the relation 
X x 2 -|- Y x 2 = R 2 , which we have seen is the condition necessary to 
make the line a tangent. Thus, given the circle x 2 -\- y 2 = 25, the 
equation of a tangent becomes xx x -\- yy x = 25. Let it be required 
to pass through (7, 1). This gives the equation of condition 
t 7x 1 -\-y x =. 25. Combining this with x* -j- y 2 — 25, expressing 
that the point sought is on the circle, we have 

x* + 625 — 350x x '+ 49xf = 25 • 
hence x x — 7x x = — 12 and x x = 4 or 3. 

The corresponding values of y x from 7x x -j- y x = 25 are — 3 and 
4, hence there are two solutions, placing the point P x at (4, — 3) 
and at (3, 4). By substitution in xx x -|- yy x = 25, we form the 
equations of two tangents to the given circle 

4x — 3i/ = 25 and 3x -f 4y = 25, 

both of which pass through the given point (7, 1). 

There are two solutions, because the given point being without 
the circle two tangents may be drawn from it to the circle. If the 
given point had been within the circle, the impossibility of drawing 
tangents to the circle through it, would have been shown by find- 
ing imaginary values for x 1 and y x . If the point were on the curve, 
there would be but one solution, equal values being found for x 1: 
and P 2 coinciding with the given point. 

Examples. — Find the tangents to x 2 -j- y 2 = 10, which pass 
through (4, 2) ; tangents to x 2 -{-y 2 = 16, passing through (4, — 4) j 
etc., -etc. 

Show that (2, 2) is within the circle x 2 -\- y 2 = 10, by the im- 



90 



THE CIRCLE. 



possibility of tangents; and that ( — 1, 3) is on the curve, by find- 
ing the single tangent that may be drawn through it. 

123. Strictly speaking, XX X -}- YY X = R 2 is not the equation 
of a tangent, but a form of the equation of the straight line, in which 
a certain relation between the arbitrary constants X x and Y x is the 
condition of tangency. 

Considering X x and Y x as two arbitrary constants, independent 
of the condition X x 2 -f- Y 2 2 ■== R 2 , the equation may represent any* 
line. We may construct it by means of its intercepts upon the 
axes (which are, in this case, perpendicular diameters). 



R 2 R 2 

An , JL n ■ . 

X/ Y x 

If Pj is a point of the circle, these values show that the intercept 
of a tangent upon a diameter is a third proportional to the corre- 
sponding co-ordinate of 

the point of contact and y v 1 

the radius. That i^, in 
the figure CR : CA : : 
CA : CT. Now suppose 
P x to be any point not 
on the curve, and join 
it with the centre. The 
equation of the diameter 



CPi is Y 



X x 



X, Art. 




65; it is therefore per- 
pendicular to XX t -j- YYj = R 2 , in which the direction ratio is 

— — 1 - Whatever the position of P x therefore, the line XX X -j- 

YY X == R 2 is perpendicular to the diameter on which P 2 is situated. 
Its distance from the centre, as found by Art. 53, is 

R 2 



l/X, 2 -f Y, 2 



* Or rather, any line not passing through the centre, which is here the 
origin ; since the equation Contains nn nbsolute term. 



TANGENT TO THE CIRCLE. 91 



But V X 2 -j- X 2 is the length CPj. Hence the distance of the 
line from the centre is a third proportional to the distance of P x , 
and the radius. Finding the point D, so that CD has this value, 
we may construct the line perpendicular to CP 2 . 

If CP X is greater than the radius, CD is less ; and if CP X is less, 
CD is greater j so that CP X X CD = R 2 . Hence if P a is without 
the circle, the equation represents a secant line as in the figure, and 
if P x is within the circle, it represents a line not cutting the circle. 
The line 

XX, -f YT, = R 2 

is called the polar of the point P x with reference ii the circle 
X 2 -j- Y 2 = R 2 , and a tangent is the polar of its point of contact. 

124. The above equation is therefore a formula for the equation 
of a tangent in terms of its co-ordinates of contact ; but more 
generally considered, it is a formula for the polar of any point, of 
which the tangent at a given point of the curve is a special case. 
The form of this equation (which is the central equation of a polar) 
shows two things independently of all we have hitherto proved. 
First, that " if one point is on the polar of another point, the latter 
is also on the polar of the first point." For suppose P 2 to be on the 
polar of P x ; we have the equation of condition 

X 2 X, + Y 2 Y X = R 2 . 

But this is also the condition that P x is on the line 

XX 2 -f- YY 2 = R 2 , 

which is the polar of P 2 . Points so related are therefore said to be 
reciprocally polar. 

Secondly, that a point polar to itself or on its own polar, is also 
on the circle of reference. For suppose V x to be such a point, we 
have the equation of condition 

X* + Y 2 = R 2 , 

which also expresses that P x is on the circle. This is the condition 
of tangency pointed out in Art. 121. 

We may now state the problem solved in Art. 122 (namely, to 
find the points of contact of tangents to x 2 -j- y 2 = 25, passing 
through (7, 1)) in the following manner: To find points which 



92 THE CIRCLE. 

shall be self-polar, with reference to x 2 -J- y 2 = 25, and at the same 
time polar to (7, 1). The solution amounts to finding points on 
the circle x 2 -J- y 2 = 25, and also on the polar of (7, 1) which is 
7x -\~ y = 25. The intersections of these are (4, — 3) and (3, 4), 
the points of tangency required (whose co-ordinates were represented 
by x x and y x in Art. 122). 

125. To find the general equation of a polar with reference to 
any circle, which of course includes the general equation of a tan- 
gent, we must substitute in 

XX, + YY X = R 2 

the general values of the central co-ordinates, both of the variable 
point P and the fixed point P x . Thus 

(x - x') {x\ - x') -f (y - y') ( 9l - y') = R 2 

is the general equation, in which x f , y' and R 2 are constants de- 
rived from the circle, and x x , y x are the co-ordinates of a given point. 
To adapt this formula to the general equation of the circle, we ex- 
pand it to 



xxi + yyt — x' (x -f x x ) —y' (y-j-yi) + x' 2 + / 2 — R 2 = 



and compare it with the expanded form of the equation of the 
circle in Art. 107. The coefficients of (jc -f- x x ) and (y -j- y^) are 
one-half those of x and y in the equation of the circle, and the last 
three terms are equivalent to the absolute term /. Or, — x' = \d, 
— y' = ie and x' 2 -f y' 2 — R 2 =/ Hence 

a»i + W\ + ? d O + x + % e (2/ + 2/x) J rf= ° 
is the general equation of a polar or tangent to the circle, 

x 2 +y 2 +dx + ey-\-f=0. 

126. It appears from these equations, that having the equation 
of a circle, the general formula for a polar with reference to that 
circle may be produced by substituting xx x for x 2 , yy x for y 2 , 
%(x -J- x x ) for x and i(y -\- y,) for y. Thus, given the circle 
x 2 -\- y 2 — 4x -j- Qy — 7 = 0, the formula for a tangent or polar is 
«*i + 2/2/i — 2 ( x + x i) + 3 (y -f y x ) — 7 = 0. 

Now the point (6, — 1) is a point of the circle, because it satis- 



TANGENT TO THE CIKCLE. 



93 



fies its equation; hence the tangent at that point is found by 
making x x = 6 and y x = — 1, which gives 

6x — y — 2x — 12 -|- 3y — 3 — 7 = 0, 



or 4tx -\- 2y 



9,9, 



or 2x 



11. 



That this line is actually tangent to the circle, may be shown by 
finding its intersection with the circle, which is in the single point 
(6,-1). 

Again, (0, 3) is a point without the circle, from which suppose 
it required to draw tangents to the circle. Making x x == and 
y x = 3, we shall have the equation of the polar line, 



3^ — 2^ + 3^ + 9 — 7 = 0, 
or Qy — 2x + 2 = 0, or x '== 3y + 1. 

The intersection of this line with the circle is in the two points 
(4, 1) and (— 2 — 1). 
These points are the 
points of contact for the 
required tangents. Fi- 
nally, the tangents at 
these points, by the same 
formula, are 




x+2y=6, andy — 2x=3, 

both of which pass 
through the given point 
(0, 3). In the annexed 
figure, the circle and 
lines of this problem are. constructed for the sake of illustration. 

Examples. — Find the equations of tangents to the circle 
x * + y 1 -f" 2x — 2y = 8, at the four points where it cuts the axes. 

Find tangents to the same circle, passing through (4, — 4) ; 
through (—1, 11). 

Determine a point on the polar of (1, 1) ; form the equation of 
its polar and show that it passes through (1, 1). 

Show generally, by means of the polar formula, the property of 
points " reciprocally polar ;" also that of " self-polar" points. 



94 THE CIRCLE. 



Length or Tangent from Given Point. 

127. To find the length, of the tangents from a given point to a 
circle, the most obvious method is to find the co-ordinates of one of 
the points of contact, and to use the formula of Art. 12. But there 
is a simpler method. For it has been shown that a tangent is per- 
pendicular to the radius at its point of contact ; hence, if we join 
both the given point, and the point of contact, with the centre, we 
shall form a right-angled triangle, from which it appears that the 
square of the tangent is the square of the distance of the point from 
the centre, diminished by the square of the radius. But the square 
of the distance of P x from the centre is Pi 

hence the square of the tangent is 

Now this expression is the result of 
substituting the co-ordinates of P x 
for x and y, in the first member of 
the equation of the circle, 

The value of the expression will be the same, if substitution is made 
in an equation of the form, x 2 -j- y 2 -J- dx -\- ey -\- f— 0, so that it 
is not necessary to find the values of x f , y' and B 2 , but only to make 
the second member zero, and the coefficients of x 2 and y 2 , unity. 
Thus, in the example of the last Art., where P x is the point (0, 3), 
and the equation of the circle is x 2 -j- y 2 — 4x -f- 6y — 7 = 0, sub- 
stitution of x = 0, y = 3 in the first member gives 20. This of 
course shows that the point is not on the circle ; but we now see 
that it also shows that the point is at such a distance that the tan- 
gent is j/20. 

128. We see then, that the expression, 

x 2 -\-y 2 -\-dx-\-ey-\-f 

gives the value of the square of the tangent to a certain circle from 
any point without it ; and the equation formed by putting this ex- 





LENGTH OF TANGENT FROM GIVEN POINT. 95 

pression equal to zero, expresses that the tangent from the point P 
to this circle is zero, or that the point P is on the circle. When 
the point P is within the circle, as in the 
annexed figure, its distance from the centre 
is less than the radius, therefore, for such 
a point, the above expression (which is 
the square of the distance, minus the 
square of the radius), is negative. For 
every point of the plane, this expression 
has a certain value, which depends upon 
the position of the point relatively to the circle, and is independent 
of the position of the origin. For the origin itself, the value of the 
expression is the absolute term • hence for any point, it is the abso- 
lute term which the equation would have if transformed to that 
point as origin* This conclusion is in accordance with the last 
paragraph of Chap. III. 

129. The value of the above expression, for any point, may 
therefore be called the absolute term for that point. Now in Arts. 
112 and 114, it is shown that the absolute term is the product of a 
secant and its external segment, or of the segments of a chord passing 
through the origin, according as it is positive or negative. We 
have now the means of readily computing the value of the absolute 
term for any point. Thus given the circle 

x * +y + 4x — 2y — 10 = -0, 

the absolute term fpr (2, 6) is found by substitution in the first 
member to be 26 ; hence 26 is the constant value of the product of 
a secant drawn through this point, and its external segment. The 
absolute term for ( — 3, 1) is — 14 j the negative sign shows that 
the point is within the circle, and 14 is the constant product of the 
segments of a chord drawn through this point, as PA . PB, in 
the figure. 

In general, let x 2 -{- y 2 -f- dx -j- ey -\-f= be the equation of the 
circle, and P, any point whose co-ordinates are x andy; and 

* Compare also the value of/ in Art. 107,/= x /2 + y /2 — R 2 , which 
shows that the absolute term is the square of the distance of the origin from 
the centre, minus the square of the radius. 



96 THE CIRCLE. 

through P let a straight line be drawn cutting the circle in A and 
B ; then 

.U^-f^ + ^+/=PAx PB, 

in which if PA and PB are measured in the same direction from 
P, they must be regarded as having the same sign and their pro- 
duct as positive ; but if they are measured in opposite directions 
they have different signs, and their product is negative. This 
equation may be regarded as including the equation of the circle 
itself, for if we put the product PA . PB equal to zero, one of the 
segments must be zero, and the point is on the circle. If we assign 
any other constant value to PA . PB, we have the equation of a 
concentric circle, because it has the effect of changing the absolute 
term only, Art. 109. In other words, for all points of a concentric 
circle this product has the same value. There is no limit to the 
positive values which may be assigned to it, but the greatest possi- 
ble negative value is — R 2 , which reduces the equation of the con- 
centric circle to an infinitesimal. 

Examples. — Given the circle x 2 -\- y 2 — 2x — \y — 20 = 0, 
what is the length of the tangent from (8, 3) ? from (6 — 2) ? from 
(0, 7) ? etc., etc. 

Find the value of PA . PB for the point (2, 2), and whether it 
is without or within the circle ) for (0, 7) ; for (5, 5) ; etc. 

Intersections of Circles. 

130. To find the intersections of two circles, it is, of course, neces- 
sary to find values of x and y, which satisfy both their equations. 
Although the rules of common algebra do not furnish a general 
method of solving two simultaneous equations of the second degree 
between x and y, yet if the equations are those of circles, they 
may be solved in the following manner : 

Let x 2 -f- y l — 2x -j- y -j- 1 = 0, and x 2 -f y 2 — \x -f Sy -f 3 = 
be the given equations. Subtracting the second from the first, we 
have an equation of first degree, 2x — 2y — 2 = 0, which may be 
combined with either of the given equations giving two solutions ; 
namely, x=l or J, y = or — \. Hence the circle represented 
by these equations intersect in (1, 0) and (£, — ?). 

As the values of x come from the solution of a quadratic equa- 



INTERSECTIONS OF CIRCLES. 97 

tion, they may be equal, in which case the circles will meet in a 
single point or touch each other; or they may be imaginary, in 
which case the circles do not intersect. 

Examples. — Find the intersections of x 2 -f y 2 — 3x -j- 4y = 10, 
and x 2 + y 2 -\- 6x-fy = 7; of x 2 + y 2 + 2x + 2y — 27 = 0, 
and x l + f — Ux — lOy — 99 = ; etc., etc. 

131. We solved the above problem by the aid of an equation of 
first degree, which was formed by combining the given equations. 
In other words, we employed the equation of a straight line, which, 
according to Art. 41, passes through the points of intersection 
sought, and then we found the intersections of this line with one 
of the circles. 

In general, let the circles in the 
figure have for their equations, 

and x 2 J r y 2 J r d'x+e f y+f'=0, 

then the equation of the straight 
line passing through the intersec- 
tions, A and B, is 

(rf_ rf > +(e _ < Oy+(/-/')=o. 

If the circles intersect, as in the figure, AB is a common chord. 
If the circles touch, A and B coincide, and the line is a common 
tangent. If the circles do not meet, the line does not cut either 
of them j but it still has a definite position, and if its equation be 
combined with each of given equations, there will result the same 
imaginary values for the co-ordinates of intersection ; and therefore 
the line is said to meet the two circles in the same imaginary 
points. 

This line, whether cutting the circles or not, is called the radical 
axis of the two circles. 

132. The intersections of two circles are thus determined by the 
intersection of a certain straight line with one of them. Therefore, 
in accordance with the language adopted in Art. 117, two circles 
are said to meet in two real, coincident or imaginary points. But 
if the circles are concentric — that is, if the equations differ only in 
their absolute terms (Art. 109), d—d' and e = e', and the equa- 




98 THE CIRCLE. 

tion of the radical axis reduces to the impossible form, a constant 
— 0. Thus, given the equations x 2 -J- y 2 — 2x -j- y -j- 1 = 0, and 
x % -\-y 2 — 2x -\- y — 5 = 0; subtracting one from the other, we 
have the impossible result 6 = 0. This, of course, indicates that 
the circles do not intersect, just as a similar result did of the lines 
x -j- y = 1 and x -j- y = 3, in Art. 40. But as parallel lines were 
said, in Art. 42, to meet at infinity, and the impossible result (of 
the form C = 0) was considered as the equation of a line at infinity, 
so we shall say that the radical axis of concentric circles is the line 
at infinity, and therefore that these circles meet at infinity. 

133. In the case of parallel lines, which, though they do not in- 
tersect, are said to intersect in points infinitely distant, we may 
interpret the expression as follows : When two straight lines are 
very near and approaching to parallelism, their point of intersec- 
tion is at a very great and increasing distance. And this increase 
goes on without limit, until when the lines are actually parallel, the 
point ceases to exist, and the distance is said to be infinitely great. 
The present case may be explained thus : When the circles are 
nearly concentric, the values of the coefficients (d — d') and (e — e!) 
in the equation of the radical axis are very small, and therefore the 
radical axis is very distant. (This is easily seen by considering its 
intercepts.) Finally, when the circles are actually concentric, the 
radical axis is infinitely distant. Now when this line was at a 
great distance from the circles its intersections with them were of 
course imaginary. But when the line is at infinity they must be 
considered infinite as well as imaginary. 

In speaking of the actual intersections of lines we should have 
to say : Straight lines may meet in one point but not more ; circles 
may meet in two points but not more. But admitting the expres- 
sions, infinite and imaginary points, we may say: Straight lines 
always meet in one point, circles always meet in two points.* 

* The real ground on which we use the expressions point at infinity, 
imaginary points, etc., in cases where intersections do not exist, is that the 
general values of the co-ordinates of intersection take these special forms 
in certain cases. If general values were found for two circles in the man- 
ner we have used in the numerical examples, d = d / and e = e / would make 
them infinite. If two equations of second degree were comhined in general 
form, there would he four solutions, two of which become infinite and imagi- 



COMBINED EQUATIONS OF CIRCLES. 99 

Combined Equations or Circles. 

134. If the equations of two circles be combined according to the 
principle explained in Art. 41, we shall have the equation, 

x 2 +y 2 +dx + ey +/+ h (x 2 + y 2 + d'x + e'y +/') = 0, 

representing a line passing through both their intersections, what- 
ever be the value of h. 

Now since the coefficients of x 2 and y 2 are the same — namely, 

1 -f- k, this equation represents a circle; except when k = — 1, 
and x 2 and y 2 disappear from the equation. In that case, the equa- 
tions are simply subtracted, one from the other, and the result is 
the equation of the radical axis. The combined equation, in which 
k is an arbitrary constant, represents therefore a series of circles 
passing through the intersections of the given circles. Thus the 
combined equation of the circles of Art. 130 is x 2 -J- y 2 — 2x -\- 
y -|-1 + h (x 2 -j- y 2 — 4x -f- 3y -f- 3) = 0. This represents a series 
of circles passing through (1, 0) and (J, — |), which we found to 
be the intersections of these circles ; and the equation 2x — 2y — 

2 = 0, which we used in finding the intersection, was but a par- 
ticular case of the combined equation. Let k = l, and we have 
(dividing by 2) x 2 -f y 2 — 3x -f 2y -j- 2 = 0. That this circle 
intersects both of the given ones in the points (1, 0) and (|, — £) 
is verified by the fact that, in connection with either of the given 
circles, it has the radical axis x — y — 1 = 0, which is the same 
as 2x — 2y — 2 = 0, the radical axis of the original equations. 

135. As h is an arbitrary constant, it may be determined by an 
equation of condition, so as to make the circle pass through a third 
point. Thus, let the circle be required to pass through (1, 2), as 
well as the points (1, 0) and (£, — J), through which it must 
pass, whatever the value of k. Substituting x = 1 and y = 2 in 

*2 + f - 2x + y + 1 + h {x 2 + y 2 - 4x + 3y + 3) = 0. 

we have the equation of condition, 6 -j- 10& = ; hence k == — -|. 
With this value of k, the equation reduces to 

nary for circles, while the other two may be real. But for concentric circles 
all four become infinite and imaginary. 



100 THE CIRCLE. 

X * _|_ y * _|_ x _ 2 y _ 2 = 0. 

We have here another proof of what was shown in Article 110 — 
namely, that in general, a circle can be found passing through three 
given points. But if the three points had been in the same straight 
line, we should have found k = — 1, and the equation would have 
reduced to that of the radical axis. 

Examples. — Give the combined equation of the circles, x 2 -f- 
y _ 3 X _j_ 4y = 10 and x 2 + y 2 + Qx + y = 7. 

Determine h so as to make the circle pass through (3, 1) ; 
(1, 4); etc., etc. 

136. When the equations combined are those of circles which 
really intersect, as in the last Article, the whole series of circles 
produced by giving different values to k, have two common points, 
which determine a common chord or radical axis. That is to say 
any two of the system of circles, 

x 2 + y 2 + dx + ey +/+ k (x 2 + y 2 +d'x + e'y +/') == 0, 

will give for their radical axis the same line, 

(d - d>) x + (e _ e ')#+ (/-/') -0. 

Again, if the original circles touch, the combined equations 
represent a system of circles, any two of which will determine the 
same radical axis, or common tangent. But even when the original 
circles do not meet, the system of circles represented by the com- 
bined equation has a common radical axis, which is a real straight 
line, and the only one, whose equation is satisfied by the common 
imaginary values of x and y, which satisfy the equations of all the 
circles. Thus if the system of equations is satisfied by the com- 
mon imaginary values, x = b ±2y — 1 and y = 3 dz^y — 1 
the common radical axis is y = 2x — 7, for this is readily seen to 
be the only equation of the form y = mx -J- 5, which these values 
of x and y will satisfy. 

Thus, whether the two points common to the system are real, 
coincident or imaginary, they determine a common radical axis. 

137. It was shown in Article 127, that the first member of the 
equation of a circle is the expression for the square of the tangent 
from any point to the circle. Hence the combined equation, 



COMBINED EQUATIONS OF CIRCLES. 



101 




x 2 + y 2 + dx + ey + /+ & (* 2 +y 2 + d'x + <y + /') = 0, 

expresses that the square of the tangent from P on one of the 
given circles, equals the square of the tangent upon the other, 
multiplied by — k. Suppose &== — 1 
(so that — k = Y), then the tangents 
from P on the two circles will be 
equal, as in the figure. But when 
k = — 1, the equation becomes that 
of the radical axis ; therefore P is a 
point of the radical axis, and that 
line is the locus of the point from 
which the tangents' on the given 
circles are equal. From this pro- 
perty, it is easy to see, that the radi- 
cal axis must be perpendicular to 

the line joining the centres of the circles, as PP' in the figure. 
Now we saw above, that the whole system of circles, produced by 
giving k different values, has a common radical axis ; therefore 
the tangents from P to any pair of circles, and consequently to 
all the circles in the system, are equal. Hence also, the centres 
of all the circles must be on the same straight line, AB ; for the 
line joining the centres of any pair of the circles is perpendicular 
to the radical axis. 

When, as in the figure, the circles do not intersect, the distance 
of any point, P, from AB is less than the common length of the 
tangents from P, and there may be found two points C and D on 
AB, at a distance from P, equal to the tangents. These points are 
infinitesimal circles belonging to the system. (See Art. 108.) If 
their equations be combined, and a proper value be given to k, we 
may form the equation of any circle of the system ; while, if we 
make k = — 1, we shall still have the equation of PP'. Hence 
PP' is the locus of a point equidistant from C and D, and each of 
the circles is the locus of a point whose distances from C and D are 
in a constant ratio. 

Examples. — Find a point from which the tangents to the three 
circles, x 2 -f- y 2 — 2x — 6> -|- 9 — 0, x 2 + y 2 — 4x + 4y — 1 = 
and x 2 -f y 2 -j- Q x — 2y -f- 6 = 0, shall be equal. (The point is 
9 * 



102 THE CIRCLE. 

on the radical axis of the first and second circles, and also on that 
of the first and third.) 

Show generally, that the three radical axes, of three circles taken 
in pairs, meet in a point (forming the general equation of the line 
passing through the intersection of two of the radical axes by Art. 
41, we shall find that the third radical axis is such a line). 

Prove from its equation, that the radical axis of any two circles 
is perpendicular to the line joining their centres. 

What is the locus of a point from which the tangent on one of 
two given circles is double that on the other ? 

What is the result of giving a positive value to k in the com- 
bined equation of two circles ? 

Ans. A circle, every point of which is without one and within 
the other of the given circles. 

State a geometrical property of all the points of the circle corres- 
ponding to h'=l, by the help of the interpretation of the first 
member given in Art. 129 and show that its centre is midway 
between those of the given circles. (This circle is only possible 
when the given circles intersect, as in Art. 131.) 

Discuss the equation x 2 -\-y 2 -J- dx -f- ey -f-/+ h (Aa: -}- By -f" C). 

Give a general equation for the circle passing through two 
given points. 



CHAPTER V. 



THE PARABOLA. 

138. If a point move in such a manner that its distances from 
a fixed point and a fixed line shall be constantly equal, it will de- 
scribe a curve which is called a parabola. The fixed point is called 
the focus, and the fixed line the directrix. 

To find the equation of the parabola, that is, of the locus of the 
point P moving according to the above law, assume as the axis of 
X a line passing through 
the focus and perpen- 
dicular to the directrix ; 
and let the origin be the 
point Y, midway be- 
tween the focus and di- 
rectrix, and the axis of 
Y, parallel to the direc- 
trix. Denote the dis- 
tance, FB, from the 
focus to the directrix, 
by p, and draw the or- 
dinate PR, and the lines PF and PD, which by the definition of the 
parabola must be equal. By the right-angled triangle PRF, PR 2 === 
PF 2 — FR 2 ; but PF = PD = BR, and hence is represented by 
x -f- ip, and FR is x — \p ; since BV and VF are each one-half 
of BF or p. Therefore y 2 = (x -j- ip) 2 — (x — Jp) 2 , or expand- 
ing and reducing, 

y 2 = 2px. 

The form of this equation shows that the curve passes through 

the origin, and that it is symmetrical with respect to the axis of 

X, as we should expect from the manner in which the origin and 

axes were chosen. It is the simplest form of the equation of a 

103 




104 



THE PARABOLA. 



parabola \ for this reason, VX is called the axis of the curve, and Y 
the vertex. To indicate that the curve is referred to its axis and 
vertex, we may use X and Y in this equation. When the proper- 
ties of the parabola are investigated geometrically, the lines VR 
and PR are denned as the abscissa and ordinate of P. We may 
therefore call these lines the geometrical abscissa and ordinate of 
P, and 

Y 2 = 2pX, 



the geometrical equation of the parabola. 

139. A chord perpendicular to the axis of the parabola, as PP 
in the figure, is called a double ordinate. It is bisected by the axis, 
and its halves are the equal positive and negative ordinates cor- 
responding to the same abscissa. The double ordinate passing 
through the focus is called the parameter. To find its value, sub- 
stitute for X the abscissa of the 
focus, which is \p, we obtain 
Y 2 = p 2 or Y = ±z p, hence the 
parameter is 2p, or four times the 
distance from the vertex to the 
focus. In the equation, 2p, the 
value of the parameter, is the co- 
efficient of the geometrical ab- 
scissa; and the equation expresses 
that the square of an ordinate 
equals the product of the parameter into the corresponding abscissa; 
or that the ordinate of a point is a mean proportional between the 
parameter and its abscissa, AB or 2p : Y : : Y : X. 

If the abscissa increase uniformly, the ordinate likewise increases, 
but at a rate which becomes slower and slower ; nevertheless, the 
ordinate may be increased to any extent, if we allow the abscissa to 
increase without limit. The curve is unlimited in extent, like 
the lines represented by equations of the first degree, and unlike 
the circle, which is a closed curve or a line returning into itself. 
Two parabolas may be compared by means of their parameters, just 
as two circles are by means of their radii ; thus, y 2 = 5x and 
y 1 =z 2x having the parameters 5 and 2, differ just as x 2 -j- y 2 = 25 
and x 2 -\- y 2 = 4, which have respectively the same numbers of units 




THE PARABOLA. 105 

for radii. Hence parabolas are considered as differing in size 
and not in shape. The figures only represent small portions of 
parabolas taken near their vertices, and the apparent shape of the 
curve depends upon the extent of the portion drawn. Thus in the 
last two figures different extents of the same parabola are drawn, 
the parameters being the same. 

140. If the vertex of a parabola is situated, not at the origin, but 
at P', the axis of the curve being parallel to the axis of X, the geo- 
metrical abscissa and ordinate, X and Y, will be denoted by as — od 
and y — y, and the equation becomes 

O— yj = 2p(x — O. 

This equation contains two new constants, just as the equation of 
a circle in any position contains two more constants than the cen- 
tral equation. However, this is not a formula for any parabola, but 
for a parabola having a given vertex and parameter, and its axis 
parallel to the axis o/X. Thus, if the vertex is to be at (2, — 1) 
and the parameter 4, the equation of the parabola is (^ -\- l) 2 == 
4 (x — 2) or f J r 2y = 4:X — 9. 

Examples. — Give the equation of the parabola with a para- 
meter 8, and vertex at (2, 6) ; parameter 3, and vertex ( — 2, 2), 
etc., etc. 

141. If in place of 2p we put a negative quantity, the equation 
represents the same parabola, but extending from the vertex toward 
the left. Thus, y % = Sx and y 2 = — 8x are the equations of equal 
parabolas, the first extending toward the right and the second 
toward the left from the origin. For whatever values of y are given 
by a positive value of x in the first, the same values of y are given 
by a like negative value of x in the second; thus (2, Hh 4) (8, ± 8) 
are points on the first, ( — 2 ± 4) ( — 8 ± 8) are points on the 
second. The first contains no points on the left of the origin, the 
second no points on the right. 

With a given vertex and a given point on the curve, 2p may be 
determined by an equation of condition, and its sign will show in 
which direction the curve extends. In fact, since 2p is four times 
YF, its sign is the same as that of YF, and therefore shows whether 
the focus (and hence the curve) is on the right or left of the 
vertex. ^ 



106 THE PARABOLA. 

Examples. — Grive the equations of parabolas extending to the 
left, equal to y 2 = 6x, and with the vertices (2, 1), ( — 1, 3), 
(0, 6), etc., etc. 

Determine the parabola with vertex (5, 2) and passing through 
(1, 1), etc. 

Polar Equations. 

142. Using the formulae of transformation from rectangular to 
polar co-ordinates, in the geometrical equation, we have 

r 2 sin 2 6 = 2pr cos 0, 

which is satisfied by r — 0, whatever the value of 0. Dividing by 
r and reducing, we find the other value of r to be 

2p cos 



sin 2 

This is the equation of the parabola when the vertex is the pole 
and the axis of the curve the initial line. If 6 = 0°, the value of r 
is infinite. If 6 = 90°, r = ; between these values r is positive. 
When is in the second or third quadrant, r is negative because 
cos is negative, and sin 2 6 is always positive. From d = 90° to 
d = 180°, the lower half of the curve is described by negative 
values of r, and for each succeeding 180° the curve is repeated. 
The line 6 = 90°, which was the axis of Y, is tangent to the curve, 
because it makes r = ; that is, does not meet the curve except at 
the pole. 

143. The polar equation when the focus is the pole takes a more 
simple form. It may be found directly from the figure of Art. 
138: thus, PF = r and FR = x, as referred to the focus; but 
from the definition, PF = BR, or r = x -f- p. This relation. be- 
tween r and x, which expresses the definition directly, is, in fact, an 
equation of the parabola, but since x == r cos 6, the true polar equa- 
tion is r = r cos 6 -f-_p, or 

P 
r= . 

1 — cos 6 

This equation gives positive values of r, for all values of 6. be- 
cause cos 6 is always less than 1. 6 == gives r = oo . 6 = 90° 



POLAR EQUATIONS. 107 

gives r =p, half the parameter or perpendicular chord through 
the focus. 6 = 180° gives r = ?p, the distance from focus to 
vertex. 

Examples. — Give the equation of the parabola whose parameter 
is 10, the focus being the pole. 

Give the values of r corresponding to 6 = 60°, = 90°, 
= 120°. 

144. The general polar equation of the parabola, as of the circle, 
is of the second degree, and each value of gives two values of r. 
The first of the two special equations we have just found, is of the 
first degree for the reason given in Art. 115, that we neglect one 
of the values of r which corresponds to every value of 6, because 
the pole was taken on the curve. In the last equation we also 
neglect one of the values of r corresponding to each value of 0; 
namely, the negative value. The double value of r may be found 
by following the regular method of transformation. Thus, the 
equation y 2 = 2px transformed to the focus (£p, 0), is y 2 = 
2px -j- jp 2 -* Transforming to polar co-ordinates, we have 

r 2 sin 2 = 2>pr cos 6 -j- p 2 . 

This quadratic for r may be solved by adding r 2 cos 2 6 to each mem- 
ber, which gives r 2 = (r cos 6 -\- p) 2 , hence 

P _ — P 



1 — cos 6 1 -|- cos 

The latter value of r is always negative, and for a given value of 0, 
is numerically the same as the positive value corresponding to 
_j_ 180°. If we give 6 all values between 0° and 360°, the whole 
curve will be described by the positive value of r. 

145. A line drawn through the focus and terminated both ways 
by the curve is a focal chord. Its length may be found, in terms 



* This is the equation of a parabola having its vertex at the point 
( — %p, 0), on the left of the origin (which is to be the focus), and might 
have been derived from the formula (y — y') 2 = 2p (z — a/). We may 
move the origin to the right by transformation, or the vertex to the left by 
the formula. 



108 



THE PARABOLA. 



of its inclination to the axis, by adding the numerical values of the 
positive and negative radii vectores corresponding to one value of 
6; thus, 

P , P __ 2 i> _ 2 P 

1 — cos 1 -f- cos 1 — cos 2 sin 2 6 

When 6 = 0°, this expression becomes infinite. When = 90°, it 
becomes 2/?, the parameter, and this is the smallest possible focal 
chord. 

Examples. — Find the focal chords having inclinations, 60°, 
45° and 30°. 

Find the chords passing through the vertex, with these inclina- 
tions. 

Show that the radius vector from the focus always exceeds the 
parallel radius vector, or chord, from the vertex. 



Secant and Tangent Lines. 

146. In finding general expressions for the intersection of a 
straight line with a parabola, we shall use for the equation of the 
straight line the form 

X = nY -f- a, 

in which, X and Y are co- 
ordinates referred to the ver- 
tex and axis of the curve, 
and n is the cotangent of 
the inclination of the line 
(Art. 46). Substituting the 
value of X, in the geomet- 
rical equation, 

Y f =2pX, 

we have 

Y 2 — 2npY=2pa, 

a quadratic of which the roots are the ordinates of the points P,P. 
Completing the square, etc., we obtain 

Y = np± \/ny + 2pa. 
The corresponding values of X, from ~K = nY -{- a, are 




SECANT AND TANGENT LINES. 109 

X = n 2 p -f a ± n\/ny -\- 2pa. 



When the radical V ' n 2 p l -J- 2pa is real, the line is a secant, as 
PP in the figure ; and the rational parts of the above values are the 
co-ordinates of M, the middle point of the chord PP. The ordi- 
nate of M is therefore np ; it is independent of a, and is the same 
for all lines in which the value of n is the same j that is, for all 
lines having the same inclination. Draw the line MPj parallel to 
the axis ; it contains all the points whose ordinates are np, hence it 
is the locus of the middle point M, when the line PP moves with 
a constant inclination; or it bisects a system of parallel chords. Such 
a line is called a diameter, therefore the diameters of a parabola 
are parallel to the axis. 

147. If the radical part of the values of X and Y is zero, the 
line is a tangent, the points P,P coinciding. Hence the condition 
of tangency is n 2 p 2 -f- 2pa = 0. When the constants, n and a, 
which determine the position of the line, are so related as to satisfy 
this equation, the line is tangent to the parabola. Thus, if the 
parameter is 8, or p = 4, the line x = 2y — 8, in which n = 2 
and a = — 8, is a tangent, as may be verified by finding its inter- 
section with y 1 = Sx. 

To derive the equation of a tangent having a given inclination, 
we must determine a in terms of n, from the condition of tangency, 
and put the value found, which is a = — \n 2r p, in the equation of 
the line j thus 

X — nY — \n*p. 

The co-ordinates of the point of contact are found by giving the 
same value to a, in the co-ordinates of intersection, X and Y, or 
what is the same thing in the co-ordinates of M, because the radi- 
cal parts become zero. Let P x denote the point of contact, then 

X x =a iri*p and Y x = np. 

These values satisfy both the equation of the curve and the equa- 
tion of the tangent. 

148. If we suppose the value of n to be the same that it was for 
the secant line PP, in the figure, the ordinate of P x is the same as 
that of M, hence the extremity of the diameter is the point of con- 
tact for a tangent parallel to the chords it bisects. P x is called the 

10 



110 



THE PARABOLA. 



vertex of the diameter PjM. The axis is one of the diameters, and 
its vertex Y is called the principal vertex. A diameter may be 
constructed, so as to bisect chords drawn in a given direction, and 
then a line drawn in the same direction through its vertex, will be 
a tangent. Thus we can construct geometrically a tangent parallel 
to a given line. 

For the equation of tangent parallel to a given line, give to n 
the value it has in the given equation, and for a perpendicular 
tangent, make n the negative of the reciprocal of n in the given 
equation. Thus, given the parabola y 2 — 8a;, in which p = 4, the 
formula for a tangent becomes x = ny — 2n 2 . Let the given line 
be 3y — 4x = 20 or x = \y — 5 ; then for a parallel n = -| , hence 
x = -| y — 1-|-; for a perpendicular tangent n = — ^, hence 
x = — %y — 3-f . 

Examples. — Grive the equations of tangents to y 2 = 6x, parallel 
and perpendicular to 2x -f- by -f- 7 = 0. 

Prove that perpendicular tangents always intersect in a point on 
the directrix (the equation of the directrix is x = — \p). 

Prove that a perpendicular to a tangent, drawn through the 
focus, (Jp, 0), intersects it on the axis of Y. 



Properties of Parabola and Tangent. 

149. Let T be the point in which the tangent at Pj cuts the 
axis. The equation of the tangent gives for the intercept X = — 
%n 2 p, which is the negative 
of the value of X x the ab- 
scissa of Pj j that is, YT = 
Y.R and is measured in the 
opposite direction. Hence 
TR, which is called the sub- 
tangent, is bisected at the 
vertex. 

The line P X N, drawn per- 
pendicular to the tangent at 

the point of contact, is called a normal; and the portion of the axis. 
RN, is called the subnormal. By similar triangles TR : PjR : : 
P a R : RN, or, since we have just shown that TR = 2X 1? Y x 2 = 
2X! X RN. But Y x 2 == 2pXj because T 1 is a point of the curve. 




OBLIQUE CO-ORDINATES OF PARABOLA. Ill 

Therefore RN = p, or the subnormal is constant and equals half 
the parameter. 

150. Since VF = %p, TF == X a -f ip = \ TN, hence a perpen- 
dicular FB to the tangent bisects TP X by the similar triangles 
TFB, TNPj. But the axis of Y also bisects TP 1} and therefore 
cuts it in the same point, B. It follows that the right triangles 
FBT, FBP X are everyway equal, and the angle FP 1 T = FTP 1 ; 
that is, the tangent makes equal angles with the axis and line 
joining the focus and point of contact, or bisects the angle between 
this line and the diameter produced. 

The inclination of the tangent is therefore one-half the angle 
FPJ). Let 6 represent the inclination of the focal radius vector, 
P X F, \0 will then represent the inclination of the tangent at P x . 
Then as 6 varies from 0° to 360°, this inclination will vary at half 
the rate, and from 0° to 180°. Consequently, for two values of d 
differing by 180°, the inclinations of the tangent will differ by 90°, 
or the tangents at the tivo extremities of a focal chord are perpen- 
dicular. 

A tangent at a given point of the curve is constructed (after the 
axis is drawn) by means of the property of the subtangent. The 
parameter focus, etc., may then be found by the property of the 
subnormal. If the curve only was given, we should draw a diame- 
ter by bisecting parallel chords; and then a parallel line, bisecting a 
perpendicular chord, will be the axis. 

Examples.— Prove that P X F X FH = P X H X VF. 

Show that TN equals half the focal chord parallel to PjT. 

Oblique Co-ordinates of Parabola. 

151. Since every diameter of the parabola bisects chords parallel 
to the tangent at its vertex, the simplest oblique equations of the 
parabola will be those which express the relation between these 
semi-chords and parts of the diameter cut off. Thus, let P^and 
PiPJ a diameter and tangent at its vertex, be taken as the co-ordi- 
nate axes, then the two halves of any chord, PP (which is bisected 
at M), will be equal positive and negative ordinates, corresponding 
to P X M, as an abscissa. 

In Art. 146, we found the geometrical co-ordinates of P,P in terms 
of the constants, n and a, to be 



112 



THE PARABOLA. 




X == n 2 p -f- a ± nyrPp 2 -\- 2pcr, 

Y == np =h ~[/n 2 p 2 -\- 2pa, 

of which the rational parts 
are the co-ordinates of M, 
and the radicals are MQ 
and PQ, the differences of 
the co-ordinates of M and 
P. By the right-angled tri- 
angle, PQM, the sum of the 
squares of these quantities 
is the square of PM. Again 
P X M is the difference of the 
abscissas of M and P l5 and 
the latter was found, in Art. 

147, to be X l = \n 2 -p. Therefore, denoting P X M and PM the 
oblique co-ordinates by Xand Y, 

Y 2 = (1 + n 2 ) (ny -f 2pa) and X = \n 2 p + a. 

These equations give us values of Y and X in terms of n and a. 
Now n is constant, because the chord PP is to be always parallel to 
P x F, and n determines the direction of the line PP ; if then we can 
establish a relation between Yand X, which is independent of a, 
it will be true for every position of the point P, and therefore will 
be the oblique equation of the curve. Now, if we multiply the 
above value of Xhj 2p (1 -f- n 2 ), we shall have the value of Y 2 , 
hence the required equation is 

Y 2 =2p(l + n 2 )X. 

152. The form of this equation is the same as that of the rec- 
tangular equation first found, and expresses that the square of the 
ordinate is a mean proportional between the abscissa and the con- 
stant, 2p (1 -j- 7i 2 ), which may be called an oblique parameter. If 
we denote this quantity by 2p', the equation becomes 

Y 2 =2p'X. 

The oblique parameter is always greater than 2p, which we shall 
call the principal parameter of the parabola. The factor 1 -f- n 2 



OBLIQUE CO-ORDINATES OF PARABOLA. 113 

is the square of the cosecant of the inclination of the new axis of 
Y, because n is its cotangent. If 6 denotes this angle, 

2»' = 2p cosec 2 = -2- : 
* r sinV 

but this is the value of the focal chord making the inclination 0, 
Art. 145, therefore the parameter for any oblique ordinates is the 
parallel focal chord, as AB in the figure. The part of the diameter, 
P^, cut off by AB, or abscissa corresponding to the ordinate p\ is 
easily shown from the equation to be \p\ or one-fourth of the para- 
meter, as in the rectangular equation. The distance P X F from the 
vertex to the focus is also one-fourth of the parameter, for it is the 
same as the distance of V 1 from the directrix, or X x -j- §p = 

h(n* + l)p. 

153. We may now drop the distinction between oblique and 
rectangular co-ordinates, and regard Y 2 = 2pX as the equation of 
a parabola having the axis of X as a diameter, the axis of Y a tan- 
gent at the origin, and 2p for the parameter parallel to the axis of 
Y. For any parabola whose axis is parallel to the axis of X, let 
P' be the point of contact of a tangent parallel to the axis of Y, and 
let 2p denote the corresponding parameter, then the equation is 

(y-yy = 2p(x-x r ). 

Thus (jj -j- l) 2 = 4 (x — 2) represents a parabola, whatever the 
inclination of the co-ordinate axes ; (2, — 1) is a point of the curve 
(as easily verified) and 4 is the parameter parallel to the axis of 
Y. If the inclination of the axes is given, the principal parameter 
may be derived from this, by the relation between parameters, in 
the last Article. For instance, if the inclination is 60°, since 
sin 2 60° = -|, the principal parameter is 3. 

154. Expanding the above equation, we have 

f - %px - yy + y 2 + 2 P x f = o, 

which contains only one term of the second degree, y 2 . Hence 

Cy + D*-J-Ey-{-F = 

is the general equation of the parabola with its axis parallel to the 
axis of X, and 

f _|_ dx -f- ey + /=0 
10* 



114 THE PARABOLA. 

is the form which it takes, when we divide through by C, the co- 
efficient of y 2 . From the values of d, e and/, in a given equation, 
the values of 2p, x' and y' ) that is, the parameter and the position 
of the vertex, may be determined. For, comparing the equation 
with the expanded form, we find d = — 2p, e = — 2y, f=y' 2 + 
2px f . Hence 

2p = - d, / = - ie and ol =P=£ = f -^f = t=M. 

2p — d 4d 

Thus, the equition Sx -f- 2y 2 -f- 10 = 4y represents a parabola. 
Arranging the terms and dividing, y 2 -j- 4x — 2y -j- 5 = 0, in 
which d = 4, e = — 2 and f = 5 ; substituting which in the values 
of 2p, a/ and y, we find that the parameter is — 4, and the vertex 
is the point ( — 1, 1). The negative value of the parameter indi- 
cates that the parabola extends from the vertex to the left, as ex- 
plained in Art. 141. 

Examples. — Determine the parameter and vertex of Qx — 2y 2 = 
4y-f3; of5— f — x-\-2y-, of/-f 2x=0; etc., etc. 

Equations of Parallel Lines. 

155. Solving the equation of the parabola' as a quadratic for 
y, we obtain 

y = — ie zb V\<? — / — dx, 

two values ofy in terms of x. The rational part is the value ofy, 
and y = — \e is the equation of a Y 

diameter of the curve, because it ex- / 

presses the value of the ordinate of / p^ — """" 

M, midway between the points of the ..M^.L 

curve corresponding to the same ab- /^4 / 

scissa. The radical part is the value V >L 1 / M 

of MP, which must be added to and \ / / 

subtracted from the ordinate of M, TFfrw- 

to produce the ordinates of the points 0/ ^^ x 

P,P, on the parabola. This radical / ^^^\^ 

contains x j it is the variable part of J 

the ordinates, and for certain values 

of x will be imaginary. The value which makes it zero will be 

found to be the same as that of x f in the last Article ; it is a limit- 



EQUATIONS OF PARALLEL LINES. 115 

ing value of x, because the value of y is only possible when — dx 
(or 2px) is algebraically greater than — (\e 2 — /). 

In the figure, the parabola is so drawn that — dx algebraically 
increases as x increases ; that is, d is negative, and the expression 
for the parameter is positive. If the parabola extended from P' 
toward the left, d would be positive and the parameter negative. 

156. If d = 0, the term containing x disappears from the equa- 
tion, and from the radical in the values of y. Hence the equation 
y 1 -\- ey -j-/= expresses that y has one of the two constant 
values 

y = — }e ± VW-^f. 

If now \e 2 >/, these values are real, and the equation is equivalent 
to two equations of the form y = b, therefore it represents two 
straight lines parallel to the axis of X. If \e 2 =/, the two values 
of y become identical, and the equation gives but one value of y, 
y = — ?e, for every value of x ; but as the two parallel lines in 
this case become one, it is called the equation of two coincident 
lines. Finally, if \e 2 <^f, the values of y are imaginary, and the 
equation is said to represent two imaginary parallel lines; for, 
though satisfied by no real points, the two imaginary values of y 
are constant. Therefore the equation of the parabola, when we 
admit zero among the possible values of the constants, may repre- 
sent two parallel, coincident or imaginary lines. 

Examples. — What lines does y 2 -|- \y = represent ? 2y 2 -f- 
4y+2 = 0? y 2 = 9? 4y— y — 5 = 0? 

157. The constant values of y, in the last Article, are also the 
intercepts of y 2 -j- dx -f- ey -\-/= on the axis of Y, as seen by 
making x = in the variable values of Art. 155. Let the para- 
bola cut the axis in the two points, B and D ; that is to say, sup- 
pose the intercepts real, or \e 2 — f (the quantity under the radical 
sign) positive. The position of the points B and D (which depend 
upon the intercepts) and that of the line P'M (which depends upon 
«/ or — \e) is not affected by any change in the value of d. But 
from the values of x' and 2p, which are 

e 2 4^ 

2p = — d, x' = -, 

it appears that as we diminish d to zero, the parameter decreases 



116 THE PARABOLA. 

without limit, and the distance of the vertex increases without limit. 
Since \e 2 — /is positive, the numerator of x' is positive, and its 
sign is the same as that of d, or opposite to that of the parameter. 
In the figure 2p is positive and x' negative, so that all values of x, 
algebraically greater than x f , give real values of y. Finally, when 
d=0, the parameter becomes zero, the vertex P' disappears, be- 
cause x r becomes infinite; and all values of x give real values of^, 
equal to the intercepts. The curve is then said to vanish into the 
parallel lines passing through B and D. 

158. If \e 2 — f were negative instead of positive, the curve 
would not cut the axis of Y, because the intercepts would be 
imaginary. In that case x' would have the same sign as 2p, or 
would be positive for a parabola extending toward the right, like 
that of the figure. So that when d = and x' becomes infinite, 
there are no values of x algebraically greater than x f , and hence 
no values which give real values of y. The curve, therefore, van- 
ishes into imaginary lines. 

If \e 2 — /= 0, the intercepts are real and equal, and x' = ; 
that is, the curve touches the axis of Y, and the vertex P' is a 
fixed point on that axis. In this case, real points occur only on the 
right of the axis when d is negative, and only on the left, when d 

is positive. But when d = 0, x' takes the form -, the position of 

the vertex becomes indeterminate, and every value of x gives equal 
values of y. The curve, therefore, vanishes into a pair of lines coin- 
cident with y = — \e. 

In general, as the parameter of a parabola changes sign, passing 
through the value zero, x' changes sign, passing through the value 
infinity, and the vertex reappears on the other side of the axis of Y. 
But in this last case, the vertex is on the axis, except in the vanish- 
ing case, when it is anywhere on the line y = — \e. 

159. An equation of the general form 

(y + D* + Ey + F=:0, 

and which will be satisfied by three given points, can always be 
found; because three equations of condition will determine the 
ratios of the four coefficients, or the values of the constants in 

y 2 .+ dx + ey+f=0, 



EQUATIONS OF PARALLEL LINES. 117 

We shall, in this way, generally determine a parabola passing 
through the three given points. Thus, let the points be (3, 1), 
(2, — 2) and ( — 1,5). Assuming the above form, the equations 
of condition are 

l_j_3rf-f- c+/=0, 

4+2^-2 e +/=0, 

and 25— rf+5e+/=0, 

from which, eliminating /, we have 

3 _ d — 3e = and 24 — 4d -f 4e = 0; 

and finally, <f = 5i, e = — t,/= — 16. The required parabola, 
therefore, is y 2 -j- §\x — \y — 16 = 0, or 4y 2 -f- 2\x — 3y — 
64 = 0, which is satisfied by each of the given points. 

If two of the given points have the same ordinate, we shall find 
d= 0, and the equation will represent a pair of lines parallel to the 
axis of X. If all three ordinates are equal, the equations will be 
found insufficient to determine e and/; for the points being on one 
straight line parallel to the axis of X, this and any line parallel to 
it will constitute a pair satisfying the conditions. If the three 
points are in one straight line, not parallel to the axis of X, the 
equations of condition will be found to be contradictory, and no 
equation of the form assumed can be found. But, even in this case, 
an equation of the general form, fulfilling the conditions, can be 
found; for let C = 0, then Dx -\- ~Ey -{- F = will with proper 
values of the coefficients represent the straight line on which the 
points are situated. 

Therefore the general equation of the parabola above becomes 
the equation of two straight lines when D = 0, and becomes that 
of a single straight line when C = 0.* 

Examples. — Determine the equation, for the points (2, — 1), 
(1, 0) and (3, 2); for (1, 1), (—1, 5) and the origin; for (1, 2), 
(0, 5) and (3, — 4) ; for (2, 0), (2, 3) and (2, 1). 

Grive the value of the parameter, etc., in each case. 

* When C = 0, the equation cannot be put in the form y 2 + dx -j- ey -f- 
f = ; just as when B = 0, Ax -\- ~By -f- C = cannot be put in the form 
y = mx -f- b. General expressions for 2p, x / and y / would all become infi- 
nite, for C = 0. 



118 THE PAKABOLA. 

Intersections of Parabolas. 
160. The intersection of a given straight line with a parabola 
of the form Gy 2 -j- Dx -J- Ey -f- F == 0, is most readily found by 
substituting the value of x from the straight line in the equation 
of the parabola. This gives a quadratic equation fory, whose roots 
are real, equal or imaginary. Accordingly the line is said to cut 
the curve in two real, coincident or imaginary points ; that is, it 
cuts the curve in two points, touches it, or fails to meet it al- 
together. 

But if the line is parallel to the axis of X ; that is, if its equa- 
tion is of the form y = b, this method of solution cannot be used; 
we have to substitute for y in the parabola its constant value, and 
that will give an equation of the first degree for ~x. Thus, given 
2y 2 -j- 3x — 4y -f 12 ' = 0, and the line y=2,we have 8 -f 3x — 
8 -j- 12 = to determine the value of cc, which is x = — 4. As an 
equation of first degree gives but a single value of x, such a line 
cuts the curve in a single point. This must be distinguished from 
the case of two coincident points, which indicates tangency. 

Examples. — Find the intersection of x -j- y === 2, with y 2 -j- x -f- 
4y -|- 2 = ; of x-\- y = 0, with the same parabola. 

Find the intercept of the curve on the axis of X, and its inter- 
sections with x = — 2, and with y = — 2. 

161. The cutting in a single point indicates that the line is a 
diameter. Hence the form of the equation of the parabola, which 
has been discussed, shows that the diameters and axis of the curve 
are parallel to the axis of X. In like manner, an equation of the 

form 

Ax 2 + Dx -j- Ey -f F = 

represents a parabola, cut in a single point by every line of the 
form x = a j that is, having its diameters and axis, parallel to the 
axis of Y. These two forms represent parabolas with particular 
directions of the axis ; the equation of a parabola having its axis 
in any direction, when combined with the equation of a straight 
line, would generally give two points, real, coincident or imaginary; 
but for lines in a certain direction, there will be but one point. 

It must be remembered that y 2 -j- dx -f- ey -\-f= is a parabola 
already fulfilling one condition, and that in Art. 159 it was shown 



INTERSECTIONS OF PARABOLAS. 119 

that besides that condition it may be made to fulfil three others ; 
namely, to pass through three given points. It will be shown in 
Chap. VIII., that in general a parabola may be made to fulfil four 
conditions, or pass through four given points. One of the con- 
ditions in Art. 159 is that the axis have a certain direction. 

162. Parabolas whose axes have the same direction may be called 
parallel parabolas. Parabolas evidently may intersect in four 
points, but parallel parabolas intersect in only two points. For, 
taking the axis of X parallel to the axes of the curves, their equa- 
tions will be of the forms y 2 -j- dx -j- ey -\-f= and y 2 -j- d'x -f- 
e'y -(-/' = 0. The equation formed by combining these is 

which, since it contains no term of the second degree, except y 2 , 
generally represents a parabola parallel to the given ones, and 
passing through all their points of intersection. But when we 
make k == — 1 ; that is, when we combine the equations by simple 
subtraction, the result is of the first degree and represents a straight 
line, which can only cut either parabola in two points. 

This straight line is analogous to the radical axis of two circles, 
and its equation may be used to find the points of intersection ; 
thus, given y 2 -f Sx — 3y -f 10 == and y 2 -f 4x -f 6 = 0, sub- 
tracting, we have 4x — 3y -f- 4 = 0. Substituting 4x = 3y — 4 
in y 2 -f- 4tx -j- 6 = 0, y 2 -f- Sy -j- 2=0. This quadratic gives 
y = — 2 or y = — 1, and finding the corresponding values of x 
from \x = Sy — 4, the two points of intersection are ( — 2 J, — 2) 
and ( — If, — 1), These points will be found to satisfy both the 
given equations. 

Examples. — Find the intersections of y 2 -\- 4x — 2y — 18 = 
and y 2 — ±x-\-2y = 0', of y 2 -f 4x -f- 2y -f 6 == 0, with each of 
them, and verify the results. 

163. If d = d f , the given parabolas are equal, as well as parallel, 
for the parameters are equal, and the signs of the parameters being 
the same, they extend in the same direction. In this case the equa- 
tion of the straight line, which -is 

(d-d') x+(e-e')y +/-/' = 0, 
takes the form y = b. Therefore by Art. 160, it can only cut 



120 THE PARABOLA. 

either parabola in one point. Hence, parallel and equal para- 
bolas extending in the same direction intersect in but one point. 

If at the same time e = e\ so that the given equations differ only 
in their absolute terms, the straight line becomes the line at infinity, 
and there is ra>* intersection, as in the case of concentric circles, 
and universally, of the loci of equations differing only in the abso- 
lute term. 

164. Another method of finding the intersections is to combine 
the equations so as to eliminate x, which gives the equation of a 
pair of straight lines, as shown in Art. 156. Each of these lines is 
of the form y = b, and intersects either of the given parabolas in a 
single point. For example, take y 2 -f- Sx — % + 10 = and 
y 2 -f- 4tx -J- 6 = 0, the parabolas whose intersections were found 
in Art. 162. Subtracting the first from twice the second gives 
f + %y '"+ 2 = 0. 

We thus obtain at once the quadratic which was solved in that 
Art., giving the two values of y ; namely, y = — 2 and y = — 1, 
which are the equations of the two parallel lines in question. 

Parabolas Passing through Fixed Points. 

165. The equation of combination 

represents a parabola fulfilling three conditions, and capable of ful- 
filling a fourth. The first of the three conditions fulfilled is that 
it is parallel to the given parabolas, because it is included in the 
general form Cy 2 -j- Dx -f- Ey -f- F = 0. The other two are, that 
it passes through two fixed points — namely, those real, coincident 
or imaginary points, in which the given parabolas intersect. The 
fourth condition will determine k j this condition may be that it 
pass through a third fixed point, as in Art. 159, or that the para- 
bola have a given parameter. The general expression for 2p is 

* The intersections which disappear in the above cases are said to be 
" at infinity." Thus, one of the intersections of a parabola and a line parallel 
to its axis, or of equal parabolas, is at infinity ; and when the axes coincide 
both of them are at infinity. 



PARABOLAS PASSING THROUGH FIXED POINTS. 121 

__D_ d + kd! 
P ~ ~~C _ """" 1 + *' 

to which, in general, we may give any value and so determine the 
value of k. Thus, given the parabolas y 1 -j- 4x -j- 6 = and 
y 2 -j- 8a; — 3^ -(- 10 = 0, and required that the new parabola have 
a parameter 6, and extend toward the left (that is, 2p = — 6) ; we 

have — 6 = — , whence k = 1. The required parabola is 

X —J— rC 

2tf-\-\2x — Zy+ 16 = 0. 

166. This method of determining k may be regarded as includ- 
ing the cases in which the equation is made to represent a single 
straight line, and a pair of parallel lines. For k = — 1 makes the 

expression for 2p infinite, and k = — — , makes it zero. Thus, the 

parabola vanishes into a single straight line, when the parameter is 
increased without limit. It is easy to see that it approximates to a 
single line when the parameter is very large, just as it approximates 
to parallel lines when it is very small. 

In case d = d'. the expression for 2p reduces to — d, whatever 
the value of k j that is, the parameter of the parabola represented 
by the combined equation is constant, when the given parabolas are 
equal. In Art. 163, it was shown that equal parabolas intersect in 
only one point, therefore the series of parabolas, on this supposition, 
are parallel and equal parabolas passing through one fixed point. 
By an equation of condition we can find one of the series pass- 
ing also through a given point. 

167. Except in the above case, 
a certain value of k will give a 
single line, and another a pair of 
parallel lines. The parallel lines 
are real, coincident or imaginary, 
according as the two points com- 
mon to the system are real, coin- 
cident or imaginary. In the 
figure the parabolas are supposed 
to touch. The common tangent, 

AB, is the single line, and CD, parallel to the axes of the curves, 
11 F 




122 THE PARABOLA. 

is the pair of coincident lines. Though CD, regarded as a single 
line, cuts the parabolas, yet as a double line it meets them in two 
coincident points, and therefore fulfils the algebraic condition of 
tangency. In fact, in the equation of any two coincident lines 
every value of x gives equal values of y, therefore it fulfils this con- 
dition with respect to every line. The parabola represented by 
the equation of combination, in this case, fulfils the two con- 
ditions of tangency to a fixed line at a fixed point. 

168. If we combine the equation of a pair of lines parallel to the 
axis of X with that of a single oblique line, thus, 

f + <& +/+ * (Ax -f By + C) = 0, 

we still have the equation of a parabola passing through two fixed 
points, whatever the value of k. The two points are those in which 
Ax -f- By -j- C = cuts the two lines represented by y 2 -J- ey -\- 
f= 0, and k may be determined by an equation of condition, so 
that the parabola shall pass through a third fixed point. This 
formula furnishes us another method of finding the parabola pass- 
ing through three given points. Thus, take the first two of the 
points in the example solved in Art. 159; namely, (3, 1) and 
(2 — 2). The lines parallel to the axis of X, passing through 
these points, are y — 1 = and y -f- 2 ='0, the equation of the 
pair is, by Art. 81, (y — 1) (y + 2) = ory 2 -fy — 2 = 0. 
The equation of the straight line is, by the formula of Art. 64, 
y — 1 = 3 (x — 3) or y — 3x -j- 8 = 0, hence 

f + y - 2 + h (y - Zx + 8) = 0, 

is the equation of a parabola passing through these two points. 
Now, if the parabola is required to pass through ( — 1, 5) also, the 
equation of condition is 28 -f- 16k = or k = — J, hence y 2 -f- 
y — 2 — £ (y — 3x -f 8) = or 4y 2 — 3y + 21x — 64 = 0, 
is the required parabola, the same that was found by the other 
method. 

169. In like manner, we may find the general equation of the 
parabola touching a given line at a given point. Thus, required 
the parabola touching the line y = x -{- 2, at the point (1, 3), 
which is a point of the line ; we combine with the equation of the 



EQUATIONS OF THE TANGENT. 123 

line, the equation of a pair of coincident lines passing through the 
point. Hence the equation 

(y-3y+k(y-x-2) = 0, 

in which k may be determined by another condition ; for instance, 
that the parabola have a given parameter. In this equation the 
parameter is k; in the general equation of Art. 168, it is — kA. 

Of course, all the parabolas found have had axes parallel to the 
axis of X ; but, from the principle of combined equations, the curve 
would still pass through two fixed points, if the parallel lines had 
any direction ; and it will hereafter be proved that it will be a 
parabola having its axis in that direction. 

Examples. — Solve the examples under Art. 159, by the above 
method. 

Find parabolas with axes parallel to the axis of X and passing 
through (1, 1) and (1 — 1); 1st, having 8 for parameter; 2d, 
passing through the origin ; 3d, making the intercept x = 2. 

Find a parabola touching x=y at (1, 1), and passing through 
(2, 3) ; a parabola tangent to the axis of Y at (0, 2), of the same 
size and direction as y 2 -|- 4x — y == 0. 

What is represented by y 2 -\- dx -f- ey -\- f -\- k (y — b) = ? 

Ans. A series of parallel and equal parabolas passing through a 
fixed point. 

Give the general equations (having one arbitrary constant, &) of 
the parabola whose parameter is 8, and which passes through (2, 1). 

What does Cy 2 -f Dx -f- Ey -f F = denote, when D, E and F 
are fixed, and C is regarded as an arbitrary constant (that is, re- 
garding C as taking the place of k) ? what when D only is arbi- 
trary ? and what when E only is arbitrary ? 

Equations of the Tangent. 

170. Since Y 2 = 2pX represents a parabola, even when the axes 
of co-ordinates are oblique, the algebraic expressions for the inter- 
sections of a straight line and parabola, in Art. 146, apply to a 
parabola referred to any diameter and the tangent at its vertex. 
Therefore, also the condition of tangency is the same, and the 
equation of a tangent and co-ordinates of the point of contact are 
of the same form — namely, 



124 THE PARABOLA. 

X = nY — Wjp, 
Xj == \n 2 p and Yj == np. 

In these equations p is half the parameter corresponding to the 
diameter to which the curve is referred, and n determines the direc- 
tion of the tangent, but is not the co-tangent of the inclination, 
except when the axes are rectangular. 

171. In case the axes are rectangular, the equation of the tan- 
gent may be expressed in terms of the inclination of a perpendicu- 
lar upon it from the origin. For, reducing it by the method of 
Art. 52, to the form x cos a -J- y sin a — p = 0, we have 

X nY 



\/l + ri> |/l + n 2 i/l + n 2 

hence cos a = - and sin a = — . 

. i/l -f- « 2 l/l 4- /t' 2 

n 2 
We have now to express the absolute term, $p — , in terms 

V\ + n 1 
of a. From the above values of sin a and cos a, we find 



COS a |/l _j_ n i 

hence the equation of the tangent in terms of a, is 

v . v • ii sin2a n 

X cos a -4- I sm a -4- W = 0. 

1 ' * COS a 

The absolute term, taken negatively, is the perpendicular from 
the vertex. By the formula of Art. 73, the perpendicular from any 
point is found by substituting its co-ordinates in the first member ; 
therefore, the perpendicular from the focus, (ip, 0), is 



, i I , sin 2 a\ _ cos 2 a 

p == \p[ cos a 4- ) = ip 

r r \ ' cos a J r c 



-f- sin 2 a ip 

cos a cos a 



This expression being of the same sign as the absolute term, the 
focus and vertex are on the same side of any tangent. For a given 
value of a, the tangent could be constructed by laying off the abso- 
lute term negatively from the vertex, or by laying off this value, 



EQUATIONS OF THE TANGENT. 125 

also negatively, from the focus. Thus, for a value of a in the first 
quadrant they must be laid off backward, or below the axis, for 
a value in the second quadrant above the axis, since cos a is then 
negative. 

Examples. — Give the values of these perpendiculars for a = 0°, 
60°, 120°, 180°, etc. 

172. If we put the above value of p' in place of the absolute 
term,* we have the equation of the tangent as referred to the focus, 

X cos a -f Y sin a + -^- == 0. 
cos a 

The angle a and the value of the perpendicular from the focus 
or negative of the absolute term, are the polar co-ordinates of the 
foot of the perpendicular. (See Art. 69.) Denoting them by 
and r, we have the following relation between them, 

\v 

r = ±— or r cos 6 = — Jp. 

cos d 

This is the polar equation of the locus of the foot of a perpendicular 
from, the focus upon a tangent. By Art. 71, it is a straight line 
perpendicular to the axis, at a distance \p to the left of the focus ; 
that is, the tangent at the principal vertex. Therefore we may 
construct a tangent with a given value of a, by drawing a line in 
the given direction, and a perpendicular to it at the point where it 
cuts the vertical tangent, as BF and BP l5 figure Art. 149. 

173. The equation of the tangent may be expressed in terms of 
the co-ordinates of its point of contact, thus : Putting X x in place 
of its value ?n 2 p, we have X = nY — X x or nY = X -f- X x ; then 
substituting for n its value from Y x = np, the result is 

yy^pcx + x,). 

This is a formula for a tangent at a given point on the parabola. 

* When we construct a line from its equation in the form x cos a -j- 
y sin a — p = 0, we have to lay off the negative of the absolute term. But, 
since the formula for p / gives the absolute term itself, as the perpendicular 
from the origin, we make this transformation by putting the value of p' 
directly for the absolute term. 
11* 



126 THE PARABOLA. 

Thus, the tangent to y 1 = 8x, at the point (2. 4), which we find to 
be on the curve, is 4y = 4 (x -\- 2) or y = x -f- 2. 

It must be remembered that this equation, like the correspond- 
ing equation for the tangent to a circle, is not the equation of a 
tangent, nor is P x a point of the line, unless it is also a point of the 
curve. That is, 

Y 1 » = 2pX l 

is the condition of the line's tangency, and this condition also ex- 
presses that P x is on the line. 

174. If we call the straight line represented by this equation 
the polar of the point P l5 with respect to the curve Y 2 = 2pX, the 
equation 

Y 2 Y X =p (X 2 + XO 

may be regarded as expressing, either that P 2 is on the polar of P x , 
or that P x is on the polar of P 2 . Therefore such points are said to 
be reciprocally polar. A self-polar point, or one on its own polar, 
is also on the curve, and its polar is a tangent. The problem of 
finding the equations of tangents from a given point to the curve 
is, therefore, equivalent to finding the self-polar points (or points 
of the curve) which are polar to the given point. These points are 
the intersections of the polar of the given 

point with the curve. Thus, let the line ^^^ 

P 2 P 3 be the polar of P 1} then tangents to ^Jf^^ 

the curve at P 2 and P 3 will pass through ^^^/ 

P v For example, the polar of ( — 1, 2), p i^'7v*7m 

with respect to the parabola y 2 = \2x, is >/ 

2y = 6 {x — 1) or 3x = 3 -f- y. The p \. 

intersections of this line with the curve ^v^^ 

are (3, 6) and (£, — 2), and the polars ^^^^ 

of these points, namely, Qy = 6 (x -f- 3) 

or y — x -f 3, and — 2y = 6 (x -f- £) or — y = 3x -f- 1, are tan- 
gents to the curve and pass through the given point ( — 1, 2). 

175. Since in the formula for the polar of P l5 

the coefficient of X is p, half the parameter, the equation of a polar 
cannot take the form y — b, nor the impossible form C = ; there- 



EQUATIONS OF THE TANGENT. 127 

fore every point has a polar, and the polar cannot be parallel to the 
axis. The polar of a point on the axis of X, that is, on the diame- 
ter to which the curve is referred, found by making Y 1 = 0, is 
=p (X -f- X x ) or X = — X 1; which represents a line parallel 
to the axis of Y, and cutting the diameter in a point at the same 
distance from the vertex, as the point P l5 but on the opposite side. 
We can, therefore, construct the polar of P x , in the figure, by 
drawing a diameter through it, and then laying off" YM = PiY, 
and drawing a line parallel to the tangent at V. Therefore, P 2 P 3 
is a double ordinate to this diameter, and the tangents at the ex- 
tremities of any double ordinate meet in the diameter. 

If P x were on the right of Y, the polar would cut the diameter 
produced, and would not cut the curve, therefore no tangents could 
be drawn through a point within the curve. The polar of a point 
on the axis is perpendicular to it, and the polar of the focus is the 
directrix. Hence if P x is on the directrix, it is reciprocally polar 
to the focus, or its polar passes through the focus. In that case, 
P 2 P 3 would be a focal chord, and therefore the tangents at the ex- 
tremities of a focal chord meet on the directrix. 

Examples. — Prove by the equation of the tangent line, that the 
tangents at the extremities of a double ordinate meet on the diame- 
ter which bisects it, and that the intercepted part of the diameter, 
as P X M in the figure, is bisected at Y. 

Prove by considering the equation of a polar as referred to any 
diameter, that points situated on any other diameter have parallel 
polars. 

176. The general equation of a tangent or polar, when the axis 
of the curve is parallel to the axis of X, is found by substituting 
for X,X X , etc. (which are co-ordinates as referred to a point on the 
curve), the differences (x — x'), (x x — x'), etc., which denote the 
same quantities. Hence 

(y — y ) Oi — f) —P O + xi — 2x') 

is the formula for a polar or tangent with respect to 

(y— y f y = 2 P (x-x f y 

Expanding it, we have 

yyx—}/ (y -f-yi) —p O + *i) + y' 2 + 2px' = o. 



128 THE PARABOLA. 

Finally, introducing the constants d, e and /, with the same values 
as in Art. 154, where the expanded equation of the parabola is 
simplified, we have 

W\ + KO* + x i) + ? e O ± &i) +/= o, 

for the tangent or polar with respect to 

Therefore, given the equation of a parabola, we find the formula 
for the polar in the same way as for the equation of the circle, by 
putting yy x in place of y 2 , and \ (x -J- a^), £ (y -j- y x ) in place of x 
and y . Thus, for the polar with respect to y 1 -\-2x — 6y -j- 15 = 0, 
the formula is ^ — |— as —J— ac x — 3 (y -f- 3/1) -j- 15 = 0. Now, sup- 
pose we require the equations of tangents to this parabola through 
the point (1, 2). By the formula, the polar of this point is 2y -|- 
x-\-l — Zy — 6 + 15 = or x = y — 10. This line will be 
found to cut the curve in ( — 5, 5) and ( — 11, — 1). The polars 
of these points by the same formula are 2y -\- x = 5 and x — 4y -j- 
7 = 0, which are tangents to the parabola and pass through the 
given point (1, 2). 

Examples. — Give the polar formula for y 2 — 4x -\- 7y -\- 10 = 0. 

Find the equations of tangents to this curve, passing through 
(1, — 1) ; through the origin ; etc., etc. 

Prove, from the general formula, the properties of " reciprocally 
polar" and " self-polar" points. 

Give the general equation of the polar to the origin, and show 
that it cuts the curve (and therefore the origin is without the curve), 
only when f is positive. 



CHAPTER VI. 



THE ELLIPSE. 




177. Ir a point move in such a manner that the sum of its dis- 
tances from two fixed points is constant, it will describe a curve 
called an ellipse ; the fixed points are called the foci. 

To find the equation of the ellipse, take for the axis of X the 
straight line passing through the foci, 
F and F', in the figure; and for the 
origin a point midway between them ; 
and let the axes be rectangular. De- 
note the distance from either focus to -A 
the origin by c, and the constant sum 
of the distances PF and PF by 2A. 
Let r and / represent these distances, 
or the focal distances of the point P ; 
then, by right triangles, 

r 2 = y 2 -j- (c -f x) 2 and r' 2 =f+( c — x) 2 . 

These are the geometrical relations between the focal distances and 
co-ordinates of P, and we have, besides, the equation, 

. r -f / = 2A, 

expressing the definition of the curve. Between the three equa- 
tions, we have now to eliminate the variables, r and /, so as to 
obtain an equation between x, y and the constants. To do this we 
must find expressions for r and / in terms of x and y. To avoid 
radicals, divide the difference of the squares r 2 — r n = 4cx, by the 

2cx 
sum r -j- / = 2 A, which gives the difference r — / = — . Com- 
bining this with r -j- V = 2A, 



A I CX 

r = A -\ 

^ A 



and 



/ = A — 



129 



130 THE ELLIPSE. 

Substituting the value of r in the first equation, or of / in the 
second, the middle terms of the expanded squares disappear, and 
we have 

A 2 -| = y 2 -J- c 2 -j- x 2 or y 2 -j x 2 = A 2 — c 2 . 

A. .A. 

178. This is the equation of the ellipse in terms of the constants 
A and c. It will take a more convenient form, if we introduce 
another constant in place of A 2 — c ? . Since A is necessarily greater 
than c, this quantity is always positive, and may therefore be de- 
noted by B 2 ; making the substitution and dividing by B 2 , 

A 2 ^B 2 

Now, if we make y = 0, we have x\ = A 2 , x = ± A ; that is, 
the curve cuts the axis of X at the distance A, to the right and 
left of the origin. In like manner, x = gives y = ± B, or the 
curve cuts the axis of Y at the distance B, above and below the 
origin. The parts of the axes A'A and B'B intercepted by the 
curve are called the major and the minor axis, and OA, OB whose 
lengths are A and B are the semi-axes. The point 0, in which the 
axes meet, is called the centre of the ellipse ; and the points A, B, A! 

and B', vertices of the major and the minor axis. The ratio — , or dis- 
tance of the focus from the centre divided by the major semi-axis, 
is called the eccentricity of the ellipse, and is denoted by e. Using 
this constant, the expressions for the focal distances of the point 
become 

r = A -j- ex and / = A — ex. 

The focal distances of the vertex A, whose abscissa is A, are A -f- c 
and A — c; the focal distances of B, whose abscissa is zero, are 
each equal to A. 

179. Since the centre is the point to which the ellipse is most 
readily referred, we shall use X and Y, in the manner explained 
in Art. 96, as central co-ordinates; then (clearing of fractions), 



A 2 Y 2 -f B 2 X 2 == A 2 B 2 



FORM OF THE ELLIPSE. 



131 



is the central equation of the ellipse, which we shall use in investi- 
gating its form, and 

A 2 (y — y'f -f B 2 (x — x') 2 = A 2 B 2 

is the equation of the ellipse with centre at P', and axes parallel to 
the axes of co-ordinates. 

In finding the central equation we took the major axis (that on 
which the foci are situated) as the axis of X, therefore A in the 
equation is supposed greater than B, and the foci are the points 
( — c, 0), (c, 0), in which c is determined by 



= A 2 



B 2 . 



If A = B, c = 0, and both foci coincide with the centre ; in which 
case, the point P is at a constant distance from the centre, and de- 
scribes a circle : accordingly we find that under this supposition, 
the equations represent circles ; that is, the ellipse with equal axes 
is a circle. 

Form of the Ellipse. 
180. The central equation may be put in the form, 

P = J(A + X)(A-X). 



X are 



Draw the ordinate PR, of any point • then A -f- X and A 
the segments A'R and AR, into 
which it divides the major axis, 
and the equation expresses that 
" the square of a perpendicular 
has a constant ratio to the pro- 
duct of the segments into which 
it divides the axis." Construct 
a circle on the major axis as 
diameter. In Art. 106, we saw 
that the square of the ordinate 
P'R of the circle is equal to the 
product of these segments ; there- 
fore Y 2 : P'R 2 : : B 2 : A 2 , or the ordinates of the ellipse and this circle, 
corresponding to the same abscissa, are in the constant ratio, B to 




132 THE ELLIPSE. 

A. This ratio (of the minor to the major axis) therefore deter- 
mines the shape of the ellipse, while the length of the major axis 
determines its size. There are thus two respects in which ellipses 
differ from one another, while circles and parabolas differ only in 
one respect, that of size. 

In a similar manner, we may show that the abscissas of the ellipse 
and the circle on the minor axis, corresponding to the same ordi- 
nate, are in the constant ratio A : B, and the property, above proved, 
of the square of a perpendicular, is true for both axes. Hence, 
if the ordinates of a circle be all increased or reduced in a given 
ratio, the resulting curve will be an ellipse ; or if the point P, in the 
figure, move in a perpendicular to the diameter of the circle, so 
that PR is in a constant ratio to P'R, it will describe an ellipse. 

181. The ordinate corresponding to either focus is found by 
making X = ± c, or X 2 == c 2 , which gives A 2 Y 2 == B*, since 

B 2 

A 2 — c 2 = B 2 ; hence Y = — , or the ordinate required is a third 

A 
proportional to the major and minor semi-axes. The double ordi- 
nate passing through the focus is called the parameter, and is de- 
noted by 2p, as in the parabola ; since 2 A : 2B : : 2B : 2p, it is a 
third proportional to the axes. The product of the segments into 
which the ordinate p divides the axis is (A — c) ( A -f- c) = 
A 2 — c 2 = B 2 ; hence the corresponding ordinate in the circle equals 
B. The figure indicates a method of finding a focus, from this 
property, which is evidently equivalent to making BF == AC. This 
construction also shows that^? : B : : B : A, and that B is a geomet- 
rical mean between A'F and AF. 

Polar Equations of the Ellipse. 

182. By the formulae of transformation to polar co-ordinates, the 
central equation of the ellipse becomes 

A 2 B 2 



A 2 sin 2 + B 2 cos 2 



This equation gives equal positive and negative values of r for each 
value of ; therefore every chord passing through the centre is 
there bisected. Such a chord is called a diameter. Putting for 
B 2 in the denominator its value A 2 — c 2 , we have 



POLAR EQUATIONS OF THE ELLIPSE. 133 

2 _ A 2 B 2 B 2 

~ A 2 — c 2 cos 2 ~ 1 — e 2 cos 2 

These are expressions . for the square of the semi-diameter, whose 
inclination is 0. When = 0°, we find r 2 = A 2 , which is its 
greatest possible value, and when = 90°, r 2 = B 2 , its least possi- 
ble value. All values of r, or semi-diameters, are therefore inter- 
mediate in length between A and B, the semi-axes. 

Examples. — The semi-axes being 5 and 3, find the semi-diame- 
ter inclined 30° to the major axis ; that inclined 45° ; that in- 
clined 60°. 

Prove that the sum of the squares of reciprocals of perpendicular 
semi-diameters is constant. 

183. The polar equation, when the pole is at the focus F, or left- 
hand focus, may be found thus : In the value of r, Art. 178, x is 
the abscissa of P as measured from the centre C, or CR, in the next 
figure; but in polar co-ordinates the value of CR is evidently 
r cos 6 — c. Hence r = A -j- e (r cos 6 — c). The value of r takes 
its simplest form when expressed in terms of e and^>, the eccentricity 
and parameter. Between the constants A, B, c, e and p, we have 

c B 2 

by combining B 2 = A 2 — c 2 , e = - and p = — , the following re- 

A -A. 

lations : 

c = Ac, B 2 = A 2 (1 — e 2 ), j9 = — = A (1 — e 2 ). 

A 

Making substitution of these values, the above equation becomes 
r = A -j- er cos — Ae 2 = p -f- er cos ) hence 



1 — e cos 

In this equation e determines the shape of the ellipse, and p de- 
termines its size. If we suppose e = 1, the value of r reduces to 
that which we found in Art. 143, for the parabola. Hence the 
ellipse becomes a parabola when the eccentricity becomes unity. 
If we make e = 0, it reduces to r=p, the equation of a circle 
whose radius is the semi-parameter. Hence the circle is an ellipse 
with no eccentricity. For the ellipse proper, the value of e is a 
proper fraction, because c is less than A ; that is, the eccentricity is 
12 



134 



THE ELLIPSE. 



between the limits one and zero, for which the ellipse becomes a 
parabola and a circle. 

184. The polar equation, just 
found, is equivalent to a very sim- 
ple relation between the radius 
vector and the abscissa of P. For 
r =p -j- er cos 0, or 

r =p -\- ex. 

Here x is FR, the abscissa mea- 
sured from the focus, and p is evi- 
dently the radius vector corresponding to x = 0. If we find a 

point D, on the left of F, at the distance FD = -, then p = e FD, 




and 



= e(FD-f sc) 



?DR. 



Draw the line DB perpendicular to the axis at D, then DR = PB, 
the perpendicular distance of P from this line, and we have proved 
that the distance o/P, any point of the ellipse, from the fixed point 
F, and the fixed line DB, are in the constant ratio e : 1. The fixed 
line is called the directrix. Since e <^ 1, the vertex is nearer F 
than D, and from the value of p, in the last Article, we find the 

distance of the centre from this vertex, A = — — — . 

1— e 2 

When e = 1, the vertex is midway between F and D, and the 
centre is at an infinite distance, the curve becoming a parabola. 
When e = 0, FD becomes infinite, therefore the circle has no 
directrix. 

Examples. — Determine the parameter and eccentricity of the 
ellipse whose semi-axes are 5 and 3, and give its polar equation. 

What values of r correspond to == 0° ? = 60° ? = 90° ? 

Determine the greatest and the least value of r in terms of p and e, 
and show, by the relations between the constants, that they are 
equivalent to A -j- c and A : — c. 

What value of makes r = A ? Ans. cos == e. 

185. The value of r given by the polar equation is always posi- 
tive, because e and cos are less than 1. There is also a negative 
value of r corresponding to each value of 0. Both values may be 



POLAR EQUATIONS OF THE ELLIPSE. 135 

found by transformation from the rectangular equation. The focus 
being the origin, the co-ordinates of the centre are c and 0. Sub- 
stituting these values for x' and y\ the co-ordinates of the centre, 
in A 2 (y — Z) 2 -f B 2 (x — x') 2 = A 2 B 2 , we have Ay -f BV — 
2B 2 cx == A 2 B 2 — B 2 c 2 = B 4 . Transforming to polar co-ordinates, 

r 2 (A 2 sin 2 -f B 2 cos 2 0) — 2B 2 cr cos = B\ 

If we substitute for B 2 its value A 2 (1 — e 2 ), Art. 183, the coeffi- 
cient of r 2 becomes A 2 (1 — e 2 cos 2 0) j then dividing through by 

B 2 c 

A 2 , and putting p and e for their values — and — , 

A A 



r 2 (1 — e} cos 2 6) — 2per cos 6 =_p 2 , 
hence r 2 =p 2 -j- 2per cos -j- eV cos 2 0, 

and r === ±: (jj -(- e;- cos 6). 

The upper sign gives the value of Art. 183, the lower gives 

—P 
r= , 

1 -f- e cos 
which is always negative. 

This value of r, taken with the positive sign, is the same that 
would be found for the right-hand focus. It is also the same that 
we obtain for 180° -f- 0. The arithmetical sum of the two values 
gives the length of a focal chord, 

1 — e 2 cos 2 

When e= 1, this value reduces to the expression we found for the 
parabola, Art. 145. 

186. Placing the centre at the point (A, 0), so that the origin 
is the left-hand vertex of the major axis, the rectangular equation 
is A 2 y 2 -j- B 2 x 2 — 2AB 2 x = 0, and transforming, 

r 2 (A 2 sin 2 -f B 2 cos 2 0) = 2 AB 2 r cos 0. 

Making the same substitutions of constants as above, etc., we have 

r 2 (1 — e 2 cos 2 0) == 2pr cos 0, 

which is, of course, always satisfied by r = 0, because the pole is 
a point of the curve. The other value of r is 



136 



THE ELLIPSE. 



2p cos 6 



1 — e 2 cos 2 

If e = l, this reduces to the corresponding equation for the para- 
bola, Art. 142, and if e = 0, to r = 2p cos 0, the equation of a 
circle whose radius isjp, Art. 115. 

Examples. — Find the polar equation when the pole is the lower 
vertex of the minor axis. 

Find the polar equation when the centre is at the point (c, p) j 
give the values of r corresponding to = ; d = 90 ; and that of 
6 corresponding to r = 0. 



Eccentric Angle. 

187. It is frequently desirable to express the two co-ordinates of 
a point of the ellipse, by means of a single variable. In the central 
equation 

1, 



5! 

A 2 



Y 2 
B 2 



since X < A and Y < B, there must be an angle whose cosine is 

X Y 

— and whose sine is — ; because these quantities are proper frac- 
tions, and the sum of their squares is, by the equation, unity. De- 
noting this angle by <p, we have 

X = A cos <p, Y = B sin <p. 

<p is here a variable angle, of which 
X and Y are such functions that 
they necessarily satisfy the equa- 
tion ; because, whatever the value 
of <p } cos 2 (p -j- sin 2 <p = \. Take 
any point P on the ellipse; the 
corresponding value of <p may be 
constructed thus : Prolong the or- 
dinate of P until it meets the cir- 
cle constructed on the major axis 
in P' -, draw the radius P'C, then 
P'CA, its inclination to the major axis, is the angle <p. For by the 




SECANT AND TANGENT LINES. 137 

definitions, , or — = cos P'CA : and — s= — — = sin P'CA, 

' CF A B CP' 

since by Art. 180 the ordinates in the ellipse and circle are in the 

ratio B : A. 

In the circle, tp is the same as 0, the angular co-ordinate of the 
point, when the pole is at the centre. In the ellipse it is called 
the eccentric angle of the point. 

Examples. — Grive the eccentric angles of the extremities of each 
axis. 

What is the eccentric angle of the extremity of the parameter ? 

. B 
Ans. The angle whose cosine is e, and whose sine is — . 

A 

x 2 j/ 2 
In the ellipse (- — = 1 ? what are the co-ordinates of the point 

for which <p = 60° ? of the point for which <p = 30° ? etc. 

188. If we construct a circle on the minor axis as diameter, the 
ordinate of the point P", in which P'C cuts this circle, will be 
B sin <p, which is also the ordinate of P; hence PP" is parallel to 
the major axis. From these properties it is easy to prove that if 
a line be drawn through P, parallel to P'C, it will cut the major and 
minor axes at distances from P respectively equal to B and A. 

Secant and Tangent Lines. 

189. In finding general expressions for the intersection of a 
straight line and ellipse, we shall use the form 

Y = mX -f b, 

for the straight line, and the central equation of the ellipse. X and 
Y are, therefore, co-ordinates as measured from the centre, b is the 
intercept on the minor axis, and, since the axes are rectangular, m 
is the tangent of the line's inclination to the major axis. Substi- 
tuting the value of Y in 

A 2 Y 2 -f B 2 X 2 = A 2 B 2 , 

and dividing by the coefficient of X 2 , we have the quadratic 

~^ :• A 2 m 2 + B 2 A 2 m 2 -f B 2 ' 

12* 



138 



THE ELLIPSE. 



completing the square, etc., 

— A 2 mb ± ABl/AV + B 2 — b 2 



X = 



A 2 m 2 + B 2 



and 



Y = 



B 2 b ± m ABl/A'W -f B 2 — V 
A 2 m 2 + B 2 



the values of Y being derived from Y == m X -\- b. 



When the radical i/AW + B 2 
ordinates of tho points P, P, in 
which the straight line cuts the 
ellipse ; and the rational parts 



b' z is real, these are the co- 



A 2 mb 



A 2 m 2 + B 2 



and 



Wb 



AW + B 2 




are the co-ordinates of M, the mid- 
dle point of the chord PP. Sup- 
pose now the value of m to remain 
fixed while that of b varies, both co-ordinates of M will vary, but 

Y B 2 

their ratio, which is — = — - — , is constant. Therefore the 
X mA 2 

middle points of all chords parallel to PP, in the figure, are situ- 
ated on the line MC, passing through the centre, whose equation is 

B 2 

Y = m'X, where m' stands for the constant ratio . Hence 

mA 2 

a system of parallel chords of the ellipse is bisected by a diameter. 

190. If the radical part of the values of X and Y is zero, the 

line is a tangent. Therefore the condition of tangency is b 2 — 

AW -j- B 2 or b = ± V Ahn 2 X B\ Thus, if A = 2 and B = 3, 

and m } the direction ratio is required to be 2, then b = ± 5 will 

make the line a tangent ; and y = 2.x -j- 5, y = 2x — 5 are two 

tangents having the same direction ratio. In general, substituting 

the algebraic value of b, we have 



Y = mX± VAV + B 2 , 

for the equations (as referred to the centre of the ellipse) of two 
parallel tangents. Let P, denote the point of contact of a tangent, 



SECANT AND TANGENT LINES. 139 

then substituting the above values of b in the co-ordinates of M 
(since for a tangent P,P and M coincide), 

— A 2 m B 2 

X! 5=3 - and Yi = 



± l/AV + B 2 ± l/A 2 m 2 + B 2 

In the figure, m is positive, and X x and Y 1 are of opposite signs, 
the positive value of Y x and negative of X x belong to the tangent 
for which the radical or value of h is positive, the others, to that 
in which b is negative. The points of tangency are the extremi- 
ties of the diameter which bisects chords parallel to the tangents ; 
they are sometimes called vertices of this diameter, the extremities 
of the axes being distinguished as principal vertices. 

191. Draw the diameter DD parallel to the tangents and chords; 
its equation is Y = mX. The equation of the diameter P X P X we 

B 2 

found to be Y = m'X. in which m' = , therefore these two 

mA 2 

diameters are connected by the relation 

B 2 

mm = — — 

A 2 

between their direction ratios. When this relation exists, one of the 
diameters bisects chords parallel to the other. Thus, if B = 1 and 
A = 2, mm! = — J, and the diameter Y === -|X bisects chords 
parallel to Y = — 2X. But the latter must also bisect chords 
parallel to the former, and in general, 



Y = mX and Y=m'X 



are each parallel to chords bisected by the other, or to tangents 
touching the ellipse at the vertices of the other. Such diameters 
are called conjugate diameters of the ellipse. The axes themselves 
are a pair of conjugate diameters. 

Examples. — Find tangents to the ellipse whose semi-axes are 
5 and 3 : 1st, parallel ; 2d, perpendicular to y = -2x — 4 ; and find 
their points of contact. 

Find the diameter conjugate to y = 2x, and the tangents at the 
vertices of y = 2x. 

Show that (X l5 Y,) satisfies the equation of the ellipse. 



140 THE ELLIPSE. 



Equations of the Tangent. 



192. The equation of a tangent to the ellipse at a given point 
may be expressed in terms of p, the eccentric angle of the point. 
For this purpose take the tangent 

Y = mX-\- l/A 2 m 2 -|- B 2 , 
which touches the ellipse in X x == — — , Y r 



V AW -f B 2 V A 2 m 2 -f B 2 

Since <p is the eccentric angle of P l5 X x = A cos <p and Y x =B 
B sin <p: comparing the values of X t and of Y l5 we can prove 

. B _ B cos c> 

l/A 2 m 2 + B 2 = - — , and m = — — . 

sin <p Asin^i 

Substituting these values and clearing of fractions, we have 

A sin <p. Y -|- B cos <p. X = AB, 

in which the arbitrary constant m is replaced by <p. 

Examples. — Give the equations of tangents at each of the prin- 
cipal vertices; at the points for which <p = 60°, y — 45° ) etc. 

Prove that a line touching the ellipse at the extremity of the 
parameter meets the major axis produced at D (Fig. Art. 184), and 
makes an intercept on the minor, equal to the semi-major axis. 

Verify that the point, whose eccentric angle is <p, satisfies the 
equation of the tangent. 

193. The equation of the tangent may also be expressed in terms 
of the inclination of a perpendicular. For this purpose we must 
reduce the equation to the form x cos a -f- y sin a =p, and then 
express the value of p in terms of a. Thus (Art. 52), 

Y mX ^ /A 2 m 2 -f- B 2 

l/l + m 2 l/l + m 2 'V 1 -f m 2 ' 

hence cos a = and 



l/l -f m 2 l/l -f- m 2 

From these values we see that the quantity under the radical sign, 
or square of the perpendicular from the centre, is A 2 cos 2 a -}- 
B 2 sin 2 a. Hence 



EQUATIONS OF THE TANGENT. 



141 




X cos a -f- Y sin a = V ' A. 2 cos 2 a -\- B 2 sin 2 a 

is the equation of a tangent, in 
which a is the arbitrary con- 
stant. We give the positive 
sign to the second member, so 
that a shall be the inclination of 
the perpendicular itself, and not 
of the perpendicular produced. 
The equation of the tangent 
parallel to PR is produced by 
replacing a by a -\- 180°, from which it is evident that perpendicu- 
lars on parallel tangents are equal. 

Examples. — Give the equation of the tangent when a = 0° ; 
when a = 45° ; when a = 90° ; when a = 180° ; etc. 

194. The following properties of the ellipse may be demonstrated 
by aid of the preceding equation. 

The sum of the squares of the perpendicular falling from the 
centre upon perpendicular tangents, is constant, and equals A 2 -f- B 2 . 
For the square of CR, whose inclination is a, is 

CR 2 = A 2 cos 2 a4-B 2 sin 2 a, 

and the square of CR/, perpendicular to the tangent PR', is 

CR' 2 = A 2 sin 2 a-f B 2 cos 2 «, 

because its inclination is 90° -j- a, and cos (90° -j- a) = — sin a, 
sin (90° -f- a) = cos a. This last expression is the square of the 
perpendicular on the tangent whose direction is a. Adding the 
expressions, CR 2 + CR' 2 = A 2 -f B 2 . 

Since CRPR' is a rectangle, the sum of the squares of two ad- 
jacent sides is equal to the square of the diagonal. Therefore 
CP 2 = A 2 -f- B 2 ; that is, the intersection of perpendicular tangents 
is at a constant distance from the centre ; and its locus is a circle, 
whose radius is the distance between the vertices of the axes. 

The product of the perpendiculars from the foci upon a tangent is 
constant, and equals B 2 . For the perpendiculars from the foci 
(c, 0) and ( — c, 0) are by Art. 73, 



142 



THE ELLIPSE. 



and 



c cos a — l/A 2 cos 2 a -f- B 2 sin 2 a 
— c cos a — V A 2 cos 2 a -+- B 2 sin 2 a, 



the product of which is B 2 . 

A 'perpendicular from a focus meets a tangent in a point of the 
circle described on the major axis. For RD equals c sin a. Hence 

CD 2 = CR 2 + RD 2 = A 2 cos 2 a + (B 2 -f c 2 ) sin 2 a — A 2 , 

or CD = A. That is, the locus of the point D is the circle whose 
centre is C, and radius equals A. 

Conjugate Diameters. 
195. In Art. 191 it was shown that when 

B 2 

mm = — — 

A 2 

each of the diameters of the ellipse whose direction ratios are m 
and m', bisects chords parallel to the other, and is parallel to tan- 
gents at the vertices of the other ; and such diameters were called 
conjugate. Since the axes are rectangular, m and m' are the trigono- 
metric tangents of the inclinations of the diameters, and the rela- 
tion between their directions is that the product of the tangents of 
their inclinations is negative, and equal to the square of the ratio 
of the semi-axes. 

Let CP and CP' be a pair of con- 
jugate semi-diameters, and let <p 
and <p' denote the eccentric angles 
of their vertices P and P\ The 
diameter Y = mX passes through 
P, whose co-ordinates are A cos <p 

Y 

and B sin y\ hence m = — = 



sin <p 



B 



tan <p. 



X 

Substitut- 



D 


/ ^^ 


^nP 


r 




~>sLp\ 




^\\ji 


^ 1 \ 1 


If 


c 


F'J 



A cos <p A 

ing values of m and m f in terms of <p and <p\ we have the relation 

tan <p, tan <p' = — 1, or tan <p' = — cot <p. 

* In Art. 192, m denotes the direction ratio of the tangent at the point 
whose eccentric angle is <f>, and is equivalent to m / of this Article. 



CONJUGATE DIAMETERS. 143 

Therefore the radii of the circle constructed on the major axis, 
CD and CD', whose inclinations are <p and <p' (Art v 187), are per- 
pendicular. Hence the eccentric angles of the vertices of conjugate 
diameters differ by 90°. 

196. Adding the squares of the co-ordinates of P, we shall have 
the square of the semi-diameter CP ; hence 

CP 2 = A 2 cos 2 <p -f B 2 sin 2 <p. 

In like manner, CP' 2 = A 2 cos 2 <p f -f B 2 sin 2 <p', or since <p' = <p ± 90°, 

CP' 2 = A 2 sin 2 <p -f B 2 cos 2 <p. 
Adding, CP 2 + CP' 2 = A 2 + B 2 . 

Hence the sum of the squares of conjugate semi-diameters is constant, 
and equal to A 2 -f- B 2 . 

Since in the circle the eccentric angle of a point is the same as 
the inclination of the radius, conjugate diameters of that curve 
are perpendicular; thus, CD and CD' are conjugate. Accordingly 
we found in Art. 119, that " a perpendicular from the centre bisects 
a chord." Every pair of conjugate diameters in the circle is equal, 
as well as perpendicular ; but in the ellipse, there is but one equal 
pair, and one perpendicular pair. The latter are the axes, the 
greatest and least diameters ; but in the circle any pair of perpen- 
dicular diameters may be taken as the axes. 

Examples. — In the ellipse whose semi-axes are 7 and 1, what 
is the length of the semi-diameter for whose vertex <p = 30° ? of 
its conjugate ? of the equal conjugate pair ?. 

Find in general, the semi-diameters for <p == 0°, <p = 90°, and 
find the length and eccentric angles of the equal conjugate pair. 

197. Draw the focal distances PF, PF'. By Art. 178 their 
values are r — A -j- ex and / ==■ A — ex. Therefore 

rr f =A 2 — e 2 x 2 . 

Now, in the value of CP' 2 , if we substitute for B 2 its value A 2 — c 2 , 
we have CP' 2 = A 2 — c 2 cos 2 y = L a — A z e 2 cos 2 <p, or since <p is the 
eccentric angle of P, whose abscissa is denoted above by x, 

CP' 2 = A 2 — e 2 x 2 = rr' ; 

that is, the product of the focal distances of the vertex of a diameter 
equals the square of the conjugate semi-diameter. 



144 



THE ELLIPSE. 



It follows that the square of a semi-diameter, added to the pro- 
duct of the focal distances of its vertex, equals A 2 -f- B 2 . 

198. If through the vertices of a pair of conjugate diameters 
tangents to the ellipse be drawn, they will form a parallelogram 
whose sides are equal and parallel to the conjugate diameters. The 
area of a parallelogram is equal to the product of one of two opposite 
sides by the perpendicular distance between them. Therefore the pro- 
posed parallelogram is the product of a diameter by the distance be- 
tween tangents parallel to it, or four times the product of a semi-diame- 
ter by the perpendicular from the centre upon a parallel tangent. 
Let a represent the direction of the 
tangent; then by the value of CR' 2 , 
Art. 194, the square of the perpen- 
dicular upon the tangent is 

CR' 2 = A 2 sin 2 a + B 2 cos 2 a, 

and by Art. 182, the square of CP, 
the semi-diameter whose direction 




CP 2 = 



A 2 B 2 



A 2 sin 2 a -j- B 2 cos 2 a 
multiplying, etc., we have 

4CP X CR' = 4AB = 2A X 2B; 

that is, the 'parallelogram formed by tangents at the vertices of any 
conjugate diameters is equal to the rectangle of the axes. 



Lines Bisecting the Angles of Focal Lines. 

199. For the equations of the lines 
joining any point of the ellipse to the 
foci, we make use of the formula for a 
straight line passing through two known 
points, 

X — X 

Let <p be the eccentric angle of the point 
P x on the ellipse, then the co-ordinates 
of P t are (A cos <p, B sin <p). Substituting these for x" and y", and 




LINES BISECTING THE ANGLES OF FOCAL LINES. 145 

for x' andy', the co-ordinates of F, ( — c, 0), and clearing of frac- 
tions, we have the equation of P X F, 

(A cos <p -f- c) y --= B sin <p (x -f- c). 

In like manner, we find the equation of P X F', passing through the 
focus (c, 0), 

(A cos <p — c) y = B sin <p (x — c). 

Wc have here the equations of P X F and P X F' expressed in terms of 
the constants of the ellipse and the single arbitrary constant <p. 

200. To find the equations of the lines bisecting the angles be- 
tween these lines ; we make use of the method of Art. 78 ; that is, 
we multiply the terms of each equation, throughout, by the square 
root of the sum of the squares of the coefficients of x and y, in the 
other ; and then add and subtract. In the first place, to find these 
radicals : The sums of the squares of the coefficients are 

A 2 cos 2 <p ± 2 Ac cos (p -\- c 2 -f- (A 2 — € 2 ) sin 2 <p, 

since B 2 = A 2 — r 2 . (The upper sign belongs to the first equation, 
the lower to the second.) These expressions reduce to 

A 2 ±z 2Ac cos (p -f- c 2 cos 2 <p = (A ± c cos ^>) 2 , 

therefore the radical for the first equation is A -j- c cos p, and for 
the second it is A — c cos 9?, 

We therefore multiply both members of the equation of P X F by 
A — c cos (p. The coefficient of y in the first member becomes 

(A 2 — c 2 ) cos <p -\- Ac (1 — cos 2 §3}, or B 2 cos <p -f~ Ac sin 2 <p. 

Hence the equation reduces to 

(B 2 cos d>-\- Ac sin 2 6) y = (A — c cos <b) B sin <f>. x -f- (A — c cos <f) Be sin (p. 

In like manner, the equation of PxF', multiplied by A -f- c cos <p^ 
reduces to 

(B 2 cos <p — Ac sin 2 <b) y = (A + c cos <f) B sin <p. x — (A -\- c cos <p) Be sin <p. 

Finally, adding the equations of P X F and P X F', thus prepared, 
13 G 



146 THE ELLIPSE. 

and dividing each term of the result by 2B, we have the equation 
of P X N * 

B cos <p. y = A sin <p. x — c 2 sin <p cos <p. 

Subtracting, and dividing by 2c sin <p, we have the equation of P X T. 

A sin <p. y = — B cos <p. x -j- AB. 

201. The last equation is identical with the equation of the tan- 
gent, found in Art. 192 ; therefore PjT is the tangent to the ellipse 
at the point P l5 whose eccentric angle is <p. A line perpendicular 
to a tangent at the point of contact is called a normal. Hence, a 
tangent bisects the exterior angle of the focal lines, and a normal 
bisects the interior angle. 

It is easy to show that the point P x of the ellipse satisfies both 
the equations ; and that the lines they represent are perpendicular, 
by Art. 48. 

The above equation of the normal is expressed in terms of a 
single arbitrary constant, <p, the eccentric angle of the point P x . 
It may thus be expressed in terms of the co-ordinates of P x : Mul- 
tiply both members by AB ; then, since A cos <p == x x and B sin <p = 
y Xl we have 

B 2 x,y = A 2 y x x — c 2 x x y x , 

which is the normal at a given point of the curve. 

Examples. — Find the normals to A 2 y 2 -f BV 2 = A 2 B 2 , at the 
principal vertices, and at the extremity of the parameter (c, p), ex- 
pressing the latter in terms of A and e. 

Show that all the normals of a circle pass through the centre. 

Show that the least distance from its vertex, in which a normal 
can cut the major axis, and the greatest distance from its vertex, 
in which it can cut the minor axis, are third proportionals to the 
semi-axes. 



* The absolute terms of the equations were of contrary signs, for 
A — c cos <j> and A -\- e cos <p are always positive ; they are in fact the values 
of the focal distances PiF 7 , and P X F. Therefore adding the equations 
gives the line bisecting the angle in which the origin is situated. See 
Art. 77. 






ELLIPSE REFERRED TO CONJUGATE DIAMETERS. 147 

Ellipse Referred to Conjugate Diameters. 

202. Since a diameter bisects chords parallel to its conjugate, the 
equation of the ellipse referred to conjugate diameters will express the 
relation between the semi-chords and the parts of the diameter cut 
off. To find this relation, we resume the values found in Art. 189, 
for the co-ordinates of the points P,P, in which a straight line cuts 
the ellipse, namely, 



X = 



A 2 mb ± ABl/A 2 m 2 -f B 2 — ¥ 
AW + B 2 



B 2 b ± mAB VAV -j- B 2 — b 2 

A 2 m 2 4- B 2 




The rational parts of these values are the 
co-ordinates of M, and the radical parts 
are the differences of the co-ordinates of 
M and P. Therefore CM 2 is the sum of 
the squares of the rational parts, and 
PM 2 is the sum of the squares of the 
radical parts. Let CP X be taken as the 
the axis of X, and CD as the axis of Y ; 
and let Xand Y represent CM and PM, 
the oblique co-ordinates of P, then 



_ y(A*ro 2 +B A ) (1 -f m 2 ) A 2 B 2 (A 2 m 2 -j- B 2 — b 2 ) 

~~ (AW -j- B 2 / (AW -f B 2 ) 2 

Here m is constant, and b 2 is a variable, upon which the values of 
X 2 and Y 2 depend. We have, therefore, two equations between 
three variables, from which to eliminate Z> 2 , and obtain an equation 
between X 2 and Y 2 . The result will be simplified by introducing 
new constants ; namely, the values of the semi-diameters CP X and 
CD. Denote the first by A and the second by B. To find A 2 , 
make b 2 = A 2 m 2 -j- B 2 in the value of X 2 ; because this, value of V 1 
makes Y= 0. and therefore gives the oblique abscissa correspond- 
ing to the ordinate zero. To find B' 1 put b = in the value of 
Y 2 , because b = makes X= 0. Thus, 



148 THE ELLIPSE. 

A> = AV + B< and & = ( 1 + m ') A ' B2 . 
AW + B 2 AW + B 2 

Dividing X 2 and Y 2 respectively by A 2 and B 2 we derive 

X 2 b 2 Y 2 4 6 2 

and — = 1 



A 2 AW-fB 2 B 2 AW + B 2 

Hence — + — = 1 or A 2 Y 2 + B 2 X 2 = A 2 B\ 
A 2 B 2 T 

203. This equation is of the same form as the rectangular cen- 
tral equation, A and B being the intercepts upon the oblique axes. 
Adding the above values of the squares of conjugate semi-diameters, 
we have A 2 -\- B 2 = A 2 -j- B 2 , as proved in Art. 196; and since 
m is the tangent of the inclination of B, if we substitute tan 6 for 
m, the value of B 2 will reduce to that of r 2 , in Art. 182. 

We may now drop the distinction between rectangular and oblique 
co-ordinates, and regard A 2 Y 2 -f- B 2 X 2 = A 2 B 2 as the equation of 
an ellipse, referred to any pair of conjugate diameters. From the 
form of the equation, it is evident, that if any numerical values of 
X and Y satisfy the equation, four points of the curve may be 
found having these values, either positive or negative, as co-ordi- 
nates. Thus, let a and b represent the numerical values, then the 
four points (a, b), ( — a, 5), ( — a, — b) and (a, — b) are all on the 
curve. The lines joining these points consecutively are parallel to 
the co-ordinate axes, and form a parallelogram inscribed in the 
ellipse, and the diagonals of the parallelogram are diameters of the 
ellipse. Two adjacent sides, which together subtend half the curve, 
are called supplementary chords of the ellipse. Therefore, supple- 
mentary chords are parallel to a pair of conjugate diameters. 

In the circle, every pair of supplementary chords js at right 
angles; but in the ellipse, perpendicular supplementary chords are 
parallel to the axes, because they form the only pair of rectangular 
conjugate diameters. When the centre of an ellipse is known, the 
axes may be constructed geometrically thus : Describe a circle con- 
centric with the ellipse, and cutting the ellipse in four points \ the 
line joining two opposite points will be a diameter, and the lines 
joining its extremities with one of the other points, will be supple- 



SIMILAR ELLIPSES. 149 

mentary chords, both of the circle and ellipse. Hence they are 
at right angles, and parallel to the axes of the ellipse. 



Similar Ellipses. 

204. Ellipses having their axes proportional are said to be simi- 
lar, because the ratio of the axes determines the shape of an ellipse. 
Similar ellipses have the same eccentricity, for the value of e de- 
pends upon the ratio of the axes. 

By the value of the square of a semi-diameter, Art. 182, 

B 2 



1 — e 2 cos 2 



we see that the semi-diameters of similar ellipses making the same 
angles with their axes, are proportional to the axes. Therefore, if the 
major axes of similar ellipses are parallel, all the parallel diameters 
have the same ratio. * 

Now the value of mm', the product of the tangents of the incli- 

B 2 

nations of conjugate diameters to the major axis, is — — ; it is 

A 

therefore the same for similar ellipses. Hence, the axes being 
parallel, the parallel diameters of similar ellipses have parallel 
conjugates. 

Again, if two ellipses are such that every pair of conjugate diame- 
ters in one is parallel to a pair of conjugate diameters in the other, 
they are similar; for the axes will then be parallel (since they are 
the only rectangular conjugate diameters), and the value of mm' 
will be the same, and hence the ratio of the axes is the same in the 
two ellipses. 

205. Since the equation A 2 Y 2 -j- B 2 X 2 = A 2 B 2 has now been 
shown to represent an ellipse, even when the axes are oblique, the 
reasoning of Art. 189 applies to an ellipse referred to conjugate 
diameters, and a straight line whose direction ratio is m. The 
value of m', the direction ratio of the conjugate diameter, will be 
of the same form, and we have the general relation, 

B 2 

mm == , 

A 2 ' 

13* 



150 THE ELLIPSE. 

between the direction ratios of any pair of conjugate diameters,* as 
referred to the pair whose lengths are 2A and 2B. 

Now, let the ratio A : B be the same in the equations of two ellipses, 
referred to the same or parallel axes ; then the values of m', correspond- 
ing to the same value of m, will be the same j that is, all parallel diame- 
ters in the two ellipses will have parallel conjugates. Therefore Ly the 
last Article, the ellipses will be similar and their axes will be paral- 
lel Thus, the equations 2x 2 -f 3y 2 = 6, and 2x 2 -f 3f = 24, when 

X^ ?/ 2 X^ // 2 X 2 ty 2 

reduced to the form 1- — = 1, are \- - = 1 and — = — = 1. 

A 2 B 2 3 2 12 8 

The ratio 2 :3 being the same as 12 : 8, the equations represent 

similar ellipses, whatever the inclination of the axes. 

206. Let n= -, then the central equation of the ellipse may be 
B 

written in the form, 

X* -f n 2 V = A 2 , 

which is a formula for the ellipse in terms of its intercept on the 
axis of X, and an abstract number or ratio, ?i, which determines its 
shape. The shape of the ellipse, however, depends not only upon 
the value of ?*, but also upon the inclination of the co-ordinate axes : 
that is, the angle between the conjugate diameters whose ratio is n. 
Thus, if n== ';TL, the equation reduces to 

X 2 -fY 2 - A 2 , 

which represents an ellipse referred to equal conjugate diameters,")* 
but not a circle unless the axes are rectangular. 

* If B = A, we have mm' — — 1. This is the general relation be- 
tween the direction ratios of lines parallel to the conjugate diameters of an 
ellipse making equal intercepts on the axes. When the axes are rectangu- 
lar, the lines are perpendicular. 

t The acute angle between the equal pair is less than that between any other 
conjugate diameters of an ellipse. For the tangent of the mutual inclination 

of any pair referred to the axes of the curve is . (See note to Art. 

1 + mm ' 
47.) Now raw/ is constant, and since m and m f are of opposite signs, we 
may consider m as positive. Therefore the denominator of this fraction is 
constant, and the numerator is the sum of two quantities whose product is 



AXES PARALLEL TO CONJUGATE DIAMETERS. 151 

In the rectangular equation of the ellipse, we considered B as 
less than A, because we referred the ellipse to its major axis, as 
the axis of X ; but hereafter, since A may denote any semi-diame- 
ter taken as the axis of X, we may have A < B, and n may have 
any value greater or less than one. 

If n = 0, the equation reduces to X 2 — A 2 , equivalent to X = A 
and X = — -A, the equations of two straight lines parallel to the 
axis of Y. 

If A = 0, the equation will be satisfied by no point except the 
origin ; unless at the same time n = 0, when the equation reduces 
to X 2 = 0, which represents two straight lines coincident with the 
axis of Y. 

Axes Parallel to Conjugate Diameters. 

207. If in the central equation of the ellipse, we substitute for 
the central co-ordinates, X and Y, their values (x — #'), (y — 3/), 
we shall have 

A 2 (y — y') 2 + B 2 (x — x') 2 = A 2 B 2 . 

This is the equation of an ellipse with centre at P', and having a 
pair of conjugate semi-diameters, equal to A and B, parallel re- 
spectively to the axes of X and Y. Thus, the equation of the 
ellipse whose centre is (2, — 1), with conjugate semi-diameters, 2 
and 3 units in length, parallel to the axes, is 4 (y -f- l) 2 -f- 
9 (x — 2) 2 = 36 or 9x 2 -f 4y 2 — 36x -f 8y -f 4 = 0. 

Using the central equation of the last Article, X 2 -f- n 2 Y 2 = A 2 , 
we have 



constant. The least value of such a sum occurs when the quantities are 
equal ; hence the tangent of the inclination or the acute angle between two 

conjugate diameters is least, when m= — m / = — ; that is, for the equal pair. 

A 

This acute angle is twice the angle whose tangent is — , and is therefore 

A 

the smaller in the more eccentric ellipse. The more oblique the axes, when 

n = 1, the more eccentric the ellipse ; and the ellipse X 2 + Y 2 = A 2 is 

the least eccentric possible, the inclination of the axes being given, for if n 

is not unity, the obliquity of the equal conjugate diameters is greater than 

that of the axes. 



152 THE ELLIPSE. 

(x-x') 2 -\-n 2 {y—y f ) 2 = K\ 

in which n determines the shape of the ellipse, A its size, and 
x', y' the position of its centre. Expanding the equation, 

x 2 -f n 2 y 2 — 2x'x — W\/y + x n -f- n 2 y' 2 — A 2 = 0. 

Any equation of the general form, 

Ax 2 -f- Cy 2 + Dx -f Ey -f F = 0, 

in which A and C have finite values, may be reduced to the form 

x 2 -f cy 2 -f "efc + ey -\-f= 0, 

where c, a?, e and f stand for the ratios of the coefficients. Com- 
paring this with the expanded form, we see that it is the equation 
of an ellipse, if c is put for n 2 , dfor — 2x f , e for — 2n 2 y' and /for 
x' 2 -f- n 2 y' 2 — A 2 . Since n 2 is essentially positive, c must be posi- 
tive in order that an equation of this form should represent an 
ellipse; or, in the general form, if A. and C, the coefficients of x 2 
and y 2 , have the same sign, the equation represents an ellipse. 

208. To determine an ellipse whose equation is given in the 
form x 2 -\- cy 2 -f- dx -f- ey -\- f= ; that is, to find the position of 
its centre, etc., we may compute n 2 ,x\y' and A 2 , by the relations 
between the constants, 

n % =.c, x ' = — %d, ?/ = — %- = — -, 

n 2 2c 

and A 2 == x' 2 + n 2 y' 2 —f= i I d 2 + - \ — /. 






H)- 



Thus, given the equation — 2x 2 — % ? -f- 20# — 12y — 9 = 0, 
which is in the general form ; dividing through by — 2, the co- 
efficient of x 2 , gives x 2 -|- 2y 2 — 10;c -j- 6y -f- 4£ = 0, in which c = 2, 
d= — 10, e = 6 and/=4i Therefore n 2 =2,x f = 5,y' = — li, 
and A 2 = 25. Hence the centre is the point (5, — 1£), the semi- 

diameter parallel to the axis of X is 5; and since n = — or 

A 
B = — , the conjugate semi-diameter parallel to the axis of Y 
n 

is 2*!/2. 



AXES PARALLEL TO CONJUGATE DIAMETERS. 153 

Examples. — Determine the ellipse 2x 2 -j- fy 2 + 8x — Qy -\- 
10 = 0; (3-\-x)(x — 2) = (5 — y) (2y — 1); etc. 

209. The equation Ax 2 + C# 2 -j- Dx -j- M# -f- F = contains 
all the terms belonging to the general equation of the second de- 
gree, except that containing xy, the product of the variables. If 
A = 0, or if C = 0, it reduces to the equation of a parabola with 
its diameters parallel to one or other of the co-ordinate axes. (See 
Arts. 154 and 161.) If both A = and C = 0, it ceases to be of 
the second degree, and represents a straight line. But if A and C 
have finite values of the same sign, it represents an ellipse. This 
restriction is necessary in order that n, the ratio which determines 
the shape, should be possible. 

Since x' and y' are determined from c, d and e, by expressions 
of the first degree, the centre P r can always be found ; but if the value 
found for A 2 is negative, A is imaginary, and no ellipse can be con- 
structed. In that case, the equation may be reduced to the form 
(x — x') 2 -j- n 2 (y — y') 2 =- a negative quantity, which can be sat- 
isfied by no values of x and y, because the sum of two squares is an 
essentially positive quantity. But, since the equation comes under 
the form of the equation of the ellipse, and is subject to the above 
restriction on the signs of the coefficients, it is said to represent an 
imaginary ellipse. 

210. If the value found for A 2 is zero, the equation may be re- 
duced to the form 

(x-x') 2 + n 2 (y-y>y = 0, 

an equation satisfied only by the point P'. Such an equation is 
said to represent an infinitesimal ellipse. Since the values of x' , i/ 
and n 2 are independent of/, the absolute term, we may suppose this 
term to vary, without affecting the position of the centre or the 
shape of the ellipse. Now from the value of A 2 , it appears that if 
/ is negative or zero, A is real and the ellipse real ; but if / be- 
comes positive and increases, A decreases until / = \ I d 2 -f- -], 

when A = 0, and the ellipse vanishes into a single point. Iff in- 
crease beyond this value, A becomes imaginary. 

Thus we see that the general equation of Art. 207 (like the 
general equation of the circle) includes certain equations which are 
satisfied only by a single point, and others which are satisfied by no 



154 THE ELLIPSE. 

points. Since it contains five coefficients, whose ratios c, d, e and f 
are four arbitrary constants, an equation of this form may be found 
whose locus shall pass through four given points. But the locus 
will not always be an ellipse ; and therefore the further discussion 
of the equation is deferred until we have shown what it represents 
in case the coefficients C and A have contrary signs. 

Similar and Parallel Ellipses. 

211. It must be remembered that the equation 

x * _j_ c f -f dx + ey -f/= 

does not include the equations of all ellipses, but only such as have 
a pair of conjugate diameters parallel to the co-ordinate axes. The 
centre may be in any position, and these diameters of any length ; 
thus, when the equation is in this form, the ellipse is determined 
by four quantities, as in Art. 208. It will be shown hereafter, that 
in general, five quantities or five conditions are necessary to deter- 
mine an ellipse; one condition being fulfilled by assuming the 
equation in the above form. Suppose it to be required that the 
diameters parallel to the axes shall have a given ratio, n. This 
second condition determines c, since c = n 2 ; hence we may write 
the equation, 

X * _j_ n 2y2 _J_ dx _|_ ey J r f = 0, 

where d, e and/* are arbitrary, for an ellipse fulfilling the two con- 
ditions. Two ellipses, in which the value of n is the same, may be 
called similar and parallel ellipses, because, by Art. 205, their axes, 
are parallel. 

212. For an ellipse similar and parallel to a given ellipse, n will 
be known ; and d, e and f may be determined by three equations 
of condition, so as to make the ellipse pass through three given 
points, just as a circle was found passing through three given 
points in Art. 110, and a parabola, in Art. 159. Therefore an 
ellipse, similar and parallel to a given ellipse, may be found passing 
through any three points, except when they are in the same straight 
line, as explained in Art. 159, in the case of the parabola with the 
direction of its axis determined. 

All circles are similar and parallel ellipses, because any perpen- 
dicular diameters may be taken as axes, and the ratio of diameters 



SIMILAR AND PARALLEL ELLIPSES. 155 

is always one of equality. Parallel parabolas may be regarded as 
the limiting case of similar and parallel ellipses, in which n becomes 
zero or infinite.* 

213. Every straight line may be said to cut an ellipse in two 
real, coincident or imaginary points, according to the character of 
the values of x and y, found by elimination between their equations. 
Two ellipses evidently may intersect in four points, but similar and 
parallel ellipses, can only intersect in two points. For let n be the 
ratio determining their shape, when they are referred to axes 
parallel to co-ordinate diameters of each, then their equations will 
be of the form x 2 -j- n 2 y 2 -f dx + ey -f / = and x 2 -f- n 2 y 2 -f 
d'x -j- e'y -\-f = 0. The equation formed by combining these is 

x 2 -f n 2 y 2 -\-dx-\~ey + f+k (x 2 +n 2 y 2 + d'x -f e'y -f /') == 0. 

which represents generally an ellipse similar and parallel to the 
given ellipses, because, dividing throughout by 1 -f- k, the terms 
of the second degree become x 2 -j- n 2 y 2 . The locus of the equation 
passes through all the points of intersection of the original ellipses. 
But when k = — 1 the equation reduces to 

(d-d')x+(e-e')y-{-f-f> = 0, 

an equation of the first degree, whose locus, being a straight line, 
can only intersect either ellipse in two points. Therefore the 
ellipses have but two points of intersection. In other words, we 
can eliminate at once both the terms of the second degree from the 
equations, and then having an equation of the first degree to combine 
with either, we can find the intersections. 

Examples. — Find the intersections of 2x 2 -J- 3y 2 -\- Sx — 6y — 
10 = with 2x 2 -{- 3/+ 6x — 4 = 0; of x 2 -j- y 2 -j- 2x — 2y -f 

* The equation of the ellipse referred to a diameter and tangent is 
"R 2 "R2 R2 "R2 rf. r, 

y 2 + — x 2 = 2 — x. Put — = p, then — = A and y 2 +^x 2 = 2px. 
A 2 A A A 2 A A 

B 2 p 

Let p remain constant, and A and B increase without limit, then — — — 

A 2 A 

decreases to zero, and we have the equation of the parabola. Thus the 

ellipse becomes a parabola when A and B are both infinite, and their ratio 

B . A . . , i.- 

— is zero, or — is infinite. 
A B 



156 



THE ELLIPSE. 




1 = with 2x 2 + 2y 2 — 4x — 4y-f-2 = 0; of 3x 2 + y 2 — 6x = 8 
with 2/ = 12x — 6x\ 

214. If the ellipses touch one another, the straight line will be 
a common tangent, as AB in 
the figure, the two points of in- 
tersection coinciding at P. Join 
P to the centres, and C, 
then CP and OP are semi-di- 
ameters of the two ellipses, 
conjugate to diameters parallel 
to AB. But since parallel di- 
ameters have parallel conju- 
gates, Art. 204, CP and OP 

must form a single straight line. Hence if similar and parallel 
ellipses touch, the point of contact is in a straight line with their 
centres. 

If the ellipses cut each other, the straight line AB will cut the 
ellipses in two points P,P, the extremities of a common chord, and 
if we join the centres of the ellipses with the middle point of the 
chord, we shall construct diameters conjugate to the diameters 
parallel to AB, which will form a single straight line as before. 
Hence a common chord to similar and parallel ellipses is bisected 
by the line joining their centres. 

When the ellipses do not meet, the line AB whose equation is 



(d-d')x+(e-e')y+f-f' = ^ 

still has a definite position, and is parallel to the diameters conjugate 
to OC. For the centres were found, in Art. 208, to be the points 



(-H- if 2 ) and (-^'> 



\ - ). The direction ratio of the 



line passing through these points, or the difference of their ordi- 
nates divided by the difference of their abscissas (Art. 65), is 

e e ' d $ 

. Multiplying this by , the direction ratio 

n 2 (d — d') e — e' 

1 B 2 A 2 

of AB, we have or , since n 2 was put for — in each ellipse. 

n 2 A 2 B* 

The product of the direction ratios of AB and OC, therefore, equals 



SIMILAR AND PARALLEL ELLIPSES. 157 

the product of the direction ratios of the diameter OC and its con- 
jugate in either ellipse j hence the latter is parallel to AB. 

215. We have seen that the equation formed by combining the 
equations of similar and parallel ellipses represents generally an 
ellipse similar and parallel to the given ones, and passing through 
their points of intersection. The line AB is analogous to the 
radical axis of circles, and is common to the whole system; and 
since we have just shown that this line determines the direction of 
the line joining the centres of any two of them, therefore all their 
centres are situated on the same straight line. The arbitrary con- 
stant k may be determined by an equation of condition, so as to find 
that one of the system which passes through a given point. 

If d = d! and e = e', that is, if the given equations differ only 
in the absolute term, the ellipses are concentric. The line AB 
will then be at infinity, for its equation will take the impossible form 
(see Art. 43); and since the centres coincide, OC will be indetermi- 
nate in direction. The combined equation in this case will repre- 
sent a series of similar ellipses whose axes coincide. 

Examples. — Find a similar ellipse passing through the inter- 
section of 2x 2 + 3y 2 -f- 8x — Gy — 10 = and 2x 2 + 3y 2 -f 6x — 
4 = 0, and also through (3, 1). 

Find ellipses similar and concentric with one of the above ; 1st, 
passing through (6, — 1) ; 2d, through the origin. 

What is represented by the equation 

(x - xj + n>(y-!/y + kly-y'-m(x- *')] = ? 

Ans. An ellipse whose shape is determined by n 2 , and which is 
tangent at P' to the straight y — i/ = m (x — x'~) ; for the first of 
the equations combined is satisfied only by the single point P\ 

By this formula, supposing the axes rectangular, give the equa- 
tion of a circle touching 2x = 3y — 1, at (1, 1), and passing 
through (2, 3). 

Give the equation of the locus of the centres of the ellipses re- 
presented by the above equation. 

Ans. y — y = — (x — x f ) ', for it passes through P' and 



14 



158 THE ELLIPSE. 

Give the general equation of the ellipse similar to 4y 2 -j- 2x 2 = 10, 
and tangent to 4y = x, at (4, 1) ; also the locus of its centre. 

Tangent at a Given Point. 

216. Since A 2 Y 2 + B 2 X 2 = A 2 B 2 represents an ellipse, whatever 
the inclination of the axes, the expressions found in Art. 189, for 
the intersections of Y = mX -j- b with the ellipse, are of general 
application. The condition of tangency is therefore the same, and 
the equation of the tangent and co-ordinates of its point of contact 
are of the same form. That is, 

Y = mX± l/AV + B 2 
is tangent to the ellipse at P 1? whose co-ordinates are 

X,= - A ' m and Y,= W 



:l/AW -{- W ±l/A 2 m 2 -f B 2 

The double sign shows that there are two tangents having 
the direction ratio m, and the same sign must be given to the 
radical in the values of X x and Y x as in the equation of the tangent. 
Whichever sign is taken, the constant or value of b in the equation, 

T>2 T? 2 ^T 

equals — , and m = — \ Making these substitutions, and 

Y x A 2 Y X 

clearing of fractions, we have 

A 2 YY 2 + B 2 XX 1 = A 2 B 2 , 

which is the equation of the tangent expressed in terms of the co- 
ordinates of its point of contact. 

217. Since this equation contains two constants, X x and Y 1} 
in place of m, it may represent any straight line if they are con- 
sidered arbitrary or independent; and only represents a tangent 
when they are connected by the relation 

A^ + KX^A*; 

that is, when P x is a point of the ellipse. This relation is therefore 
the condition of tangency. 

In general, the line is called the polar of the point P 1} with re- 
spect to the ellipse, as in the case of the corresponding formulae for 
the circle and parabola. The equation 



TANGENT AT A GIVEN POINT. 159 

A 2 Y 2 Y X -f B 2 X 2 X X = A 2 B 2 

then expresses either that P 2 is on the polar of P x or that T 1 is on 
the polar of P 2 , hence points connected by this relation are said to 
be reciprocally polar. A point on its own polar, or self-polar point 
is a point of the curve, and its polar is a tangent, as in Arts. 124 
and 174. 

From these properties it follows, as in the Articles referred to, that 
the points of contact for tangents passing through a given point are 
the intersections of the curve with the polar of the point. For the 
given point is on the polar of the required point (that is, the re- 
quired tangent) ; hence the required points are on the polar of the 
given point. 

218. The polar of every point on a given diameter of the ellipse 
is parallel to the conjugate diameter. For the direction ratio of 

B 2 X 

the polar is - 1 which is constant as long as the ratio of X x 

A 2 Y X 

and Y x is unchanged ; that is, while 

P x is on the diameter CP^ But 

the polar of A, the vertex of this 

diameter, is the tangent at that 

point, which has been shown to be 

parallel to the conjugate diameter. 

This is evidently true, to whatever 

conjugate diameters the lines be 

referred, for the equation of CP X passing through a given point, and 

Y, 

the origin is Y == — X, and the product of its direction ratio and 
Xi 

B 2 

that of the polar is . Accordingly, if CP X is made the axis of 

A 
X, so that Y x = 0, the equation of the polar is XX X == A 2 or 

A 2 

X = — , showing that it is parallel to the axis of Y. In this 
X, 

equation, A represents the semi-diameter CA, which was made the 

axis of X, and X x represents the distance CP X . Hence CM is a 

third proportional to CP X and CA. We may construct the polar 

of a given point by laying off this distance and drawing a parallel 

to the conjugate diameter. 




160 THE ELLIPSE. 

The chord P 2 P 3 is a double ordinate to the diameter CA, and 
since by the last Article the tangents at P 2 and P 3 pass through P x , 
we see that the tangents at the extremities of a double ordinate 
meet on the diameter produced, and make an intercept which is a 
third proportional to the abscissa and semi-diameter. Compare 
Art. 123, remembering that all the conjugate diameters of the circle 
are at right angles. 

Examples. — Prove the above property of tangents by means of 
the equation of the tangents referred to CA as axis of X. 

Show that the polar of no point can pass through the centre, and 
that the centre has no polar. 

Prove that the polar of the focus is the directrix on the same 
side of the centre ; and show that therefore the tangents at the ex- 
tremity of a focal chord meet in the directrix. 

The polar of a point without the curve is a secant ; find the con- 
dition that P x should be without the curve, by applying condition 
of secancy, Art. 189, to its polar. (Reduce the polar to the form 
Y = mX -j- b, and assume the quantity under the radical sign in 
the values of the co-ordinates of intersection to be positive.) 

219. Substituting for the central co-ordinates, X,X l5 etc., their 
values (x — a/), (x x — se'), etc., we have 

A 2 (y - y) (# -y f ) + B 2 (x - *f) (x x - x') = A 2 B 2 , 

for the polar of P l5 with respect to the ellipse whose centre is P' ; 

or dividing by B 2 . and putting n for the ratio — , 

B 

(x - x') (x x - x') + n* o —y) ( yi -y) =* A 2 . 

Expanding, the equation becomes 

xx x -f n 2 yy x — x f (x -f x x ) — »V (y + y x ) + x n + nty 2 — A 2 = 0, 

and introducing the constants, c, d, e and f with the same values as 
in Art. 207, we have 

xx x -f- cyy x -f \d (x + x x ) -f \e (y -f y x ) +f= 0, 

for the polar with respect to the ellipse, 

x* + cy*+<lx-\-ey+f=Q. 



TANGENT AT A GIVEN POINT. 161 

We therefore produce the formula for the tangent or polar with 
respect to an ellipse whose equation is given, in the same way as in 
the previous cases of the circle and parabola ; namely, by substitut- 
ing xx x for x 2 in the equation of the curve, yy x for y 2 , i (x -f- x{) 
for x and ? (y -j- y x ) for y. Thus, given the ellipse 4x 2 -J- 9y 2 — 
Sx -\-lSy — 167 = 0, the formula for a polar is Axx x -j- 9yy x — 
4x — 4:x x -f 9y -f 9y x — 167 = 0, or 

(4x, — 4>* + (%, + 9)^—4^ + 9 yi - 167 = 0. 

Examples. — Find tangents to the above ellipse passing through 
the point ( — 4, 2-J-) ; through (16, — 1). (The required tangents 
are the polars of the intersections of the curve with the polars of 
the given points.) 

Show, by the general equation, that the centre generally has no 
polar, and that the polar of no point can pass through the centre ; 
but that when the ellipse is infinitesimal, every polar passes through 
the centre. (See Art. 208 for co-ordinates of the centre.) 
14* 



CHAPTEK VII 



THE HYPERBOLA. 



220. If a point move in such a manner that the difference of its 
distances from two fixed points is constant, it will describe a curve 
called an hyperbola. The two fixed points are called the foci. 

To find the equation of the hyperbola, take for the axis of X, the 
straight line passing through 
the foci, F and F'; and for 
the axis of Y, a perpendicu- 
lar line bisecting F'F. Let 
c denote the distance from, 
either focus to the origin ; 
and 2A, the constant differ- 
ence of the lines PF' and 
PF, or / and r, the focal 
distances of the point P. 

Then, drawing the ordinate of P, we have from right-angled tri- 
angles, the geometrical relations, 

r n — y 2 -\- (x -j- c) 2 and r 2 = y 7 + (x — c) 2 ; 

and from the definition of the curve, 

/ — r = 2A. 




We have now to combine these three equations so as to derive an 
equation between x, y and the constants. From the first two we 

2c,x 
have / 2 — r 2 == 4cx; dividing by / — r = 2 A, / -f- r = — '-. 

A 
Hence 



r = — 4- A 

A^ 



and 



ex 

r = A. 

A 



162 



THE HYPERBOLA. 163 

Finally, substituting the value of / in the first equation, or that 
of r in the second, we have 

c 2 x 2 c 2 — A 2 

1- A 2 = y 2 -4- x 2 4- c 2 or x 2 — y 2 = c 2 — A 2 . 

A 2 A 2 * 

221. This is the equation of the hyperbola in terms of the con- 
stants A and c. Like the corresponding equation of the ellipse, 
it will take .a more convenient form when we introduce another 
constant. Since in this case c is necessarily greater than A, 
c 2 — A 2 is a positive quantity, and may be denoted by B 2 . Making 
the substitution, and dividing by B 2 , we find 

A 2 B 2 

Now if we make y = 0, we have x 2 = A 2 or x = ± A ; that is, 
the curve cuts the axis of X in two points, A and A', on the right 
and left t)f the origin. The focal distances of A are / = A -f- c 
and r = A — c, of which / exceeds r by the required difference 
2 A. But for the point A', r' is less than r; it would therefore 
seem as if that point did not satisfy the condition / — r = 2A, 
used in deducing the equation. We shall find, however, by exam- 
ining the analytical expressions for r and /, that the condition is 

satisfied by A'. Putting e in place of the ratio — , which is called 

the eccentricity of the hyperbola, the expressions of the last Article 
become 

/ = ex -J- A and r = ex — A. 

Since c > A for the hyperbola, e >> 1. Now for the point A', 
x = — A, therefore the values, both of / and of r, are negative, and 
/, which has the least numerical value, still exceeds r algebraically 
by the required difference 2 A. For every negative value of x 
numerically greater than A, we have, in like manner, negative 
values of the focal distances, of which that of r' exceeds that of 
r algebraically by the constant difference 2A. Hence the hyper- 
bola consists of two branches, one on the right and one on the left 
of the axis of Y. 

222. The part A' A, of the line joining the foci, is called the 



164 THE HYPERBOLA. 

transverse axis of the curve, A and A' are its vertices. The mid- 
dle point, 0, is called the centre. In investigating the form and 
properties of the curve we shall use X and Y, as in Chapters IV. 
and VI., to denote central co-ordinates. Then (clearing of frac- 
tions, etc.), 

A 2 Y 2 __ B *X 2 = — A 2 B 2 

is the central equation of the hyperbola, and 

A 2 O — y) 2 — B 2 (x — x') 2 = — A 2 B 2 

is the rectangular equation of the hyperbola with centre at P' and 
transverse axis parallel to the axis of X. 

The equation of the hyperbola in terms of A and c, Art. 220, 
is of the same form as that of the ellipse, Art. 177, the difference 
being only in the comparative values of A and c. But, since B 2 
is in this case put for c 2 — A 2 , whereas in the case of the ellipse 
it was put for A 2 — c 2 , the equations containing B differ from the 
corresponding ones of the ellipse in the sign prefixed to B 2 . The 
value of c is determined by the equation 

c 2 = A 2 -f B 2 . 

Whatever the values of A and B in this equation, c is a real 
quantity; therefore the above central equation always represents 
an hyperbola having its foci on the axis of X. 

Form of the Hyperbola. 

223. Making Y an explicit function of X, the central equation 
takes the form 

T& ± - ]/X 2 — A 2 - 

From this we see that there is no point of the curve having an 
abscissa numerically less than A ; for such a value of X would 
make Y imaginary. But for all values of X, positive or negative, 
numerically greater than A, equal positive and negative values of 
Y maybe found, and points of the curve constructed. Considering 
now, only positive values of X and Y, Y is an increasing function 
of X. That is, when X == A, Y = 0; anoj as X increases, Y in- 



FORM OF THE HYPERBOLA. 



165 




creases. Now, since X 2 — A 2 is less than X 2 . the radical is less 

than X, and Y < - X. 
A 

Suppose now the values of A and B to be given ; construct the 
point (A, B), by laying off 
from C a distance equal to A 
on the axis of X, and erect- 
ing a perpendicular equal to 
B. Join the point D, so 
found, with the centre, then 
the equation of CD (passing 
through this point and the 

origin) is Y = — X. As 

.A. 
the value of X increases, the difference between X and the radical 

l/X 2 — A 2 continually decreases ; for the difference of the squares 
of these quantities (which is the product of their sum and differ- 
ence) is constant, and as X increases their sum increases. Hence, 
the ordinate to the curve is always less than the ordinate to the 
line CD, corresponding to the same abscissa, but continually ap- 
proaches to it as the abscissa is increased ; or the distance between 
the straight line and curve decreases without limit, when the 
abscissa increases without limit. 

The lower portion of the same branch of the curve approaches, 

B 

in like manner, to the line CD', whose equation is Y = X ; 

and the other branch of the hyperbola is similarly situated within 
the opposite angle formed by the lines CD and CD'. The form of 
the curve is, therefore, that represented in the figure. 

224. The straight lines CD and CD', which continually approach 
the curve, but can never meet it, are called the asymptotes. The 
asymptotes make equal angles with the transverse axis. The ratio 
B : A determines this angle, and may therefore be considered as 
determining the shape of the hyperbola. The angle between the 
asymptotes, which is double this angle, also determines the shape 
of the curve. If B < A, as in the figure, the angle DCA, or in- 
clination of an asymptote is less than 45°; and DCD', the angle be- 
tween the asymptotes, is an acute angle. If B = A, DCA will be 



166 THE HYPERBOLA. 

45°, and DCD' a right angle. In this case the hyperbola is called 
equilateral or rectangular. The hyperbola of the figure may be 
considered acute, but if B were greater than A, it would be obtuse. 
Since the perpendicular sides of the triangle CAD were con- 
structed respectively equal to A and B, and c 2 = A 2 -j- B 2 , Art. 
222, the hypothenuse CD = CF the distance from the centre to the 
focus. This triangle may therefore be constructed when the values 
of A and c are given, as well as when A and B are given. From 
the trigonometric definitions, 

tan DC A = -, sec DC A = — = - = k 

A' CA A 

therefore the eccentricity also determines the shape of the curve. 
225. The ordinate corresponding to either focus is found by 

T>2 

putting X = ± c or X 2 = c 2 , which gives Y = ± — , since 

A. 
c 2 — A 2 = B 2 . This ordinate is denoted by jp, as in the cases of 
the parabola and ellipse, and the double ordinate passing through 
the focus or 2p is called the parameter. The following are the 
most useful relations between the constants A, B, c, e andp found 

c B 2 

by combining e = — , B 2 = c 2 — A 2 and p = — : 
J ° A' r A 

c = A e, B 2 = A 2 (e 2 — 1), p = A (e 2 — 1). 

After the shape of the hyperbola, or angle between the asymptotes, 
is determined by the value of e or by the ratio B : A, its size may be 
determined by A, c or p. Hyperbolas having the same asymptotes 
being regarded as of the same shape, those which approach nearest 
to the asymptotes will be the less in size. 

By the above relations, the values of the remaining constants may 
be computed when any two of them are given ; and the inclination 
of the asymptotes may be computed by the trigonometric tables. 

Examples. — Find the parameter and the eccentricity when 
A == 2 and B = 4. 

Find the value of A when p = 3 and e == 2, and show that the 
angle between the asymptotes is 120°. 

Express B 2 in terms of p and e. 

What is the eccentricity of the equilateral hyperbola ? 



POLAR EQUATIONS. 167 



Polar Equations. 

226. By the formulae of transformation, the polar equation of 
the hyperbola, when the pole is the centre, is 

— A 2 B 2 



A 2 sin 2 — B 2 cos 2 



It is evident that every value of 6 which makes this value of r 2 
positive will give equal positive and negative values of r. Hence 
a straight line drawn through the centre meeting one branch of the 
hyperbola will, if produced, meet the other branch at an equal dis- 
tance from the centre. Such a line is called a diameter, therefore 
a diameter is bisected at the centre. Putting for B 2 , in the denomi- 
nator, its value c 2 — A 2 ' we have 



— A 2 B 2 —W 

r 2 = - 



- c 2 cos 2 1 — e 2 cos 2 

Since the numerator of the last expression is negative, the value 
of r 2 will be positive, and the value of r real, only when the de- 
nominator is also negative. When = 0° this is the case, because 
e > 1 (since c > A for the hyperbola) j the result being r 2 = A 2 , 
which is its least possible value. As 6 increases, the denominator 
decreases and r 2 increases, until 6 reaches such a value that the 

denominater becomes zero, which takes place when cos 6 = - or 

e 

sec 6 = e. This value of 6 is the inclination of the asymptote, as 
shown in Art. 224, and it makes r 2 infinite. A greater value of 
6 makes r 2 negative and r imaginary, until we reach a value in the 
second quadrant for which sec = — e. which again makes r 2 in- 
finite. This value is supplementary to the former, and is therefore 
the inclination of the other asymptote. Values of between this 
and 180° make r 2 positive and r real ; therefore every line passing 
through the centre, and between either asymptote and the axis. 
cuts both branches of the curve and is a diameter. The asymptotes 
are thus the limits of the diameter, and all diameters pass through 
the angle DCD', or interior angle of the asymptotes. 



168 THE HYPERBOLA. 

227. The denominator of the above value of r 2 is of the same 
form as that which occurs in the corresponding equation for the 
ellipse. Art. 182, and the equations differ only in the sign prefixed 
to B 2 . When either of these curves is referred to a focus or vertex, 
its equation will be most conveniently expressed in terms of p and 
e, and it will then be found that the same equation will represent 
either curve, according as we suppose e greater or less than unity. 
Also, if we make e = 1, these equations will reduce to the corre- 
sponding forms for the parabola. 

These three curves are included under the general name of the 
conic sections* The size of the conic section may be regarded as 
determined by p, and its shape by e. The parabola, in which 
e = 1, is of definite shape, and is intermediate between the ellipse 
in which e < 1 and the hyperbola in which e ]> 1. The circle, 
in which e = 0, is a particular case of the ellipse ; in its equation 
R takes the place of p. 

228. To show that these curves may be expressed by a general 
rectangular equation, containing the constants p and e, we refer the 
hyperbola to the right-hand focus F. The centre is, therefore, on 
the left of the origin, at the point ( — c, 0).f Substituting its co- 
ordinates for x' and y f in A 2 (y — y') 2 — B 2 (x — x') 2 == — A 2 B 2 , 
and expanding, we have 

Ay — B V — 2B 2 cx = — A 2 B 2 -j- B 2 c 2 = B*. 

If now we divide each member by A 2 , and make use of the rela- 
tions between the constants, Art. 225, we have 

y 2 -f- (1 — e 2 ) x 2 — 2pex —p 2 } 

or x 2 -f- y 2 = (p -j- ex) 2 . 

The equation of the ellipse in Art. 185 may be reduced to the 

* This term is derived from the fact that these are the only curves which 
can be produced by the intersection of a right cone by a plane. 

f The right focus of the hyperbola must be regarded as corresponding 
to the left focus of the ellipse, so that the nearest vertex will be in each 
case on the left. Then, as the centre of the hyperbola is on the left, both 
of the focus and vertex, c and A change signs (passing through infinity) as 
we pass from the ellipse to the hyperbola. Replacing A, c and B 2 , by 
— A, — c and — B 2 , while p and z continue positive, all the equations of 
the ellipse become equations of the hyperbola. 



POLAR EQUATIONS. 169 

same form, by employing the relations between the constants in 
Art. 183. When e = i, it reduces to y 2 = 2px -\-p 2 , found for 
the parabola in Art. 144; and when e = 0, it reduces to x 2 -f- y 2 =p 2 
the equation of a circle whose radius is p. Therefore it may repre- 
sent any conic section. 

229. Since x 2 -f y 2 = r 2 , the above equation transformed to polar 
co-ordinates is r 2 = (jo -f- e r cos #) 2 or r = =t (p + er cos *0» gi ym g 

p l !l ' —p 



1 — e cos 1 -f- e cos 

the same two values of r which were found for the ellipse. But, 
since e > 1 for the hyperbola, the first value is no longer always 
positive, nor the second always negative, as in case of the ellipse. 
Thus, when 6 = 0° the first value of r is negative, and by the re- 
lations between the constants may be proved equal to — (A -f- c) : 
hence this value of gives the vertex of the left branch. As 6 in- 
creases, the negative value of r increases, describing the lower part 
of the left branch of the curve, until sec 6 = e, when it becomes 
infinite. Therefore when r has an inclination equal to that of the 
asymptote, its value passes through infinity, and becomes positive. 
As the inclination is increased, the right branch of the curve is 
described, r remaining positive until we reach the value in the 
fourth quadrant, for which again sec 6 = e, and r is infinite. Be- 
tween this value and 6 = 360°, r is negative and describes the 
upper part of the left branch. Thus, during an entire revolution, 
both branches are described ; the right by positive values of r, the 
left by negative values. 

The second value of r may, in like manner, be shown to describe 
both branches during an entire revolution; but it describes the 
right branch by negative values, and the left by positive. 

230. The first value of r, which is positive for the branch within 
which the focus is situated, might have been derived from the value 
of r, in Art. 221. Now x in r = ex — A denotes the abscissa CR 
measured from the centre, and CR is x -\- c, when x represents FR, 
the abscissa measured from the focus. Therefore r = ex-\-ec — A, 
or since c = Ae and A (e 2 — 1) =p (Art. 225), 

r =p -j- ex. 
15 H 



170 



THE HYPERBOLA. 




This is the same relation as that shown in Art. 184 to exist be- 
tween the radius vector and 
abscissa of the ellipse as re- 
ferred to its focus ; it is equi- 
valent to the first value of r, 
and applies to all conic sec- 
tions.* 

In the hyperbola, the value 
of r thus found is positive 
for the right branch, and neg- 
ative for the left, as shown in Art. 221. 

231. If the perpendicular DB be drawn, at the distance FD = - 

e 
to the left of the focus, it may be proved (as for the ellipse in Art. 
184) that the distances of a point of the curve, from a fixed point 
or focus F, and a fixed line or directrix DB, are in the constant 
ratio e:l. Thus the three conic sections may be defined by a 
common property, from which the general equation of the conic 
section, Art. 228, may easily be derived. For, the origin being at 

F, PF 2 = x 2 + y 2 and PB = DR ==-? -f x; but by the above pro- 

e 

perty PF = ePB =p -j- ex ; hence x 2 -\- y 2 = (p -j- ex) 2 . 

For the hyperbola, the point is nearer to the directrix than to the 
focus, and therefore may be found on the left of the directrix, which 
is impossible for the parabola or ellipse. 



* Since the value of r is not affected by changing the direction of the 
axes without change of origin, and since this transformation does not alter 
the absolute term, every equation of the form 

r -f- mx -j- ny =-p] 

that is, every linar equation between r, x and y, represents a conic section, 
of which the origin is a focus and the absolute term is the semi-parameter. 
Supposing the axes rectangulai*, the formulae of Art. 88 give for the equa- 
tion, when the axes are turned through the angle a, r -j- (m cos a -f- 
n sin a) x -j- (n cos a — m sin a) y — p. The term containing y disappears 

when tan a = — ' and the equation reduces to r -f- ex — p f in which 
m 

e = zb Vm 2 -f n 2 according to which of the two values (differing by 
180°) we take for a. When e is made positive, positive x is measured 
toward the nearest vertex. 



POLAR EQUATIONS. 171 

232. The length of the focal chord, in any conic section, is the 
algebraic difference between the two values of r corresponding to 
the same value of 0. Thus, subtracting the second value in Art. 
229, from the first, we have 

p — p 2p 

1 — e cos 1 -f- e cos 1 — e 2 cos 2 
In the ellipse, the expression thus obtained is always positive, be- 
cause the first value of r was always positive, and the second always 
negative, and therefore changing the sign of the second we have 
two positive quantities to add. But in the hyperbola, the first 
value is positive for the right branch, and negative for the left; and 
so is the second value after its sign is changed. (See Art. 229.) 
Therefore, for a chord which cuts a single branch, as PF produced, 
the expression is found to be positive ; but for a chord which cuts 
both branches, as the exterior part of P'F, it is negative. Chords 
parallel to either asymptote are infinite, and a chord parallel to a 
diameter (which, according to Art 226, has a less inclination to the 
axis, so that sec 2 < e 2 ) must be regarded as negative. 

Examples. — In the conic section for which p = 3 and e = 1J, 
find the focal radius vector for = 60° ) find also the focal chords 
for = 30°, = 45°, and = 60°. 

If a line drawn through the focus, parallel to an asymptote, meet 
the hyperbola in P, show that PF equals \ of the parameter, and 
that the foot of the ordinate of P is midway between the focus and 
directrix. 

Show that the distance of the directrix from the centre is — . 

e 

Prove that the distance of a point of the hyperbola from the 
focus is equal to its distance from the directrix measured on a line 
parallel to an asymptote. 

Find the distance of the centre of x 2 -f ?/ 2 == (p -f- ex) 2 from the 
focus ; and discuss its position, supposing p to be fixed, and e to 
vary from a large positive to a large negative value. (The centre 
is midway between the points in which the curve cuts the axis 
of X.) 

Find a rectangular and a polar equation, in terms of p and e, for 
the hyperbola referred to the right-hand vertex ; and show that 
they a.pply to all conic sections. 



172 



THE HYPERBOLA. 



Conjugate Hyperbolas. 

233. The equation of an hyperbola, whose foci are situated on 
the axis of Y, might be found by a process similar to that of Art. 
220, in which x would take the place of y, and y that of x. Hence 
we infer that the result would be of the same form as if we made 

x^ ?/ 2 
this interchange at once in the equation — = 1, the result 

A x> 

of the previous process. If at the same time we interchange A 
and B, we shall have 



B 2 



= 1, 



which represents an hyperbola of which B is the transverse semi- 
axis -measured on the axis of Y. Accordingly, if x = we have 
y = ± .B, but y=0 gives imaginary intercepts on the axis of X. 

x 1 y l 
This hyperbola is called the conjugate of — — — = 1. Using 

X and Y as before for central co-ordinates, we have for its central 
equation, after clearing of fractions, 

A 2 Y 2 _ B 2 X 2 = A 2 B2? 

which differs from the other equation only in the sign of the second 
member. B, which is the transverse semi-axis of the conjugate 
hyperbola, is called the conjugate semi-axis of the original hyperbola. 

234. To construct the asymptotes of the conjugate hyperbola, we 
should have to construct the same point (A, B), as in Art. 223, 
laying off from C a distance equal to.B on the axis of Y, and erect- 
ing a perpendicular equal to A. 
Hence the two hyperbolas have 
common asymptotes, which are 
the diagonals of a rectangle 
whose centre is C, and whose 
sides are equal to 2A and 2B. 
The lines AA' and BB' are 
called the axes of both hyper- 
bolas. 

The distance CD, in the figure, is equal to the distance of the 
foci of both hyperbolas from the centre, because c 2 = A 2 -f- B 2 , and 




CONJUGATE HYPEEBOLAS. 173 

is therefore the same for each. But the eccentricity of the conju- 
gate hyperbola is the secant of DCB, or the cosecant of DC A whose 
secant is e. Therefore the two hyperbolas have different shapes. 
In fact, being situated in the supplemental angles of the asymptotes, 
v one of them is acute and the other obtuse, except when the asymp- 
totes are at right angles, and then both are rectangular or equi- 
lateral. 

235. The equations of the two asymptotes are Y = — X and 

.A. 

Y = — - X, or AY — BX = and AY + BX = 0. The com- 

A ^ 

pound equation, representing at once all the points on both these 
lines, is by Art. 81, 

A 2 Y 2 _ B 2 X 2 = 

This is the equation of the asymptotes. Solving it for Y gives 

Y = ± — X equivalent to the two equations representing the 

A 
asymptotes separately. The equation of the asymptotes differs from 
the equation of either hyperbola only in having zero for its second 
member. 

236. In order to express the two co-ordinates of a point on the 

ellipse in terms of a single variable, we made use in Art. 187 of the 

trigonometric formula sin 2 -j- cos 2 = 1, because in the equation of 

the ellipse the sum of two squares is unity. For the hyperbola we 

make a similar use of the formula sec 2 — tan 2 = 1. Thus, in the 

X 2 Y 2 X Y 

equation = 1, — and - are the secant and tangent of the 

A 2 B 2 A B & 

same angle. Denoting this angle by </>, we have 

X = A sec <p and Y = B tan <f>, 

which satisfy the equation of the hyperbola whatever be the value 
of <p. 

The application of this auxiliary angle is similar to that of the 
eccentric angle in case of the ellipse, although it cannot be con- 
structed geometrically for a given point with the same facility. It 
may, however, be shown by the algebraic signs of the functions, that 
values of </> in the first quadrant correspond to points on the upper 

15* 



174 THE HYPERBOLA. 

part of the right branch ; values in the second quadrant correspond 

to the lower part of the left branch ; values in the third, to the 

upper part of the left; and values in the fourth, to the lower part 

of the right branch. 0° and 180° correspond to the vertices A 

and A' j and 90° and 270°, for which the secant and tangent are 

both infinite, may be regarded as corresponding to points at infinity 

in the directions of the asymptotes AY = BX and AY = — BX. 

For a point on the conjugate hyperbola, we may in like manner 

assume X = A tan <p and Y = B sec <p, for these values always 

. „ . . Y 2 X 2 „ 

satisfy the equation — =-l. 

J l B 2 A 2 

Examples. — What are the co-ordinates of the point for which 
<P = 45° ? 

Ans. X = A j/2, and. Y = B. The abscissa being the diagonal 
of a square constructed on the semi-transverse axis, this point may 
be constructed when the curve and its axis are given, in order to 
find a line equal to B. 

Prove that the value of 4' f° r tne extremity of the parameter is 
equal to the inclination of the asymptote.* 



Secant and Tangent Lines. 

237. To find general expressions for the intersections of a straight 
line with an .hyperbola, we substitute Y = mX-f-Sin the central 
equation, A 2 Y 2 — B 2 X 2 = — A 2 B 2 . 

Thus, (A 2 m 2 — B 2 ) X 2 -f- 2 A 2 m6X == — A 2 B 2 — A 2 6 2 

is the equation giving the abscissas of the points of intersection. 
This is generally an equation of the second degree ; but when 
A 2 m 2 — B 2 = 0, it becomes an equation of the first degree, and 
gives a single value of X. Therefore, when A 2 m 2 = B 2 , or 

m = ± — j that is, when the line is parallel to either asymptote, it 



* This angle, whose secant is e, is used as an auxiliary constant in astro- 
nomical computations. Thus ife== sec ^, B^Atan^, andp = A (e 2 — 1) = 
A tan 2 V- Similarly in the ellipse, if e = cos 0, B == A sin 0, andp = 
A sin 2 0. See example under Art. 187. 



SECANT AND TANGENT LINES. 



175 



T> 

cuts the hyperbola in a single point. If m = — , we find for the co-or- 

A 

{T> 7 v /- T>2 ^ 
1 I and Y — i\b L 
b B J I b ) 

which become infinite when 6 = 0. Therefore the asymptote itself 
does not meet the curve. 

238. Dividing the above equation by the coefficient of X 2 , and 
completing the square, 



x + 



A 2 mb 



A 2 m 2 - B 



A 2 B 2 (A 2 m 2 — B 2 ) + A 2 B 2 6 2 

(A 2 m 2 — B 2 ) 2 



Hence we have for the general values of the co-ordinates of inter- 
section, 

A 2 mb ± ABl/B 2 — A 2 m 2 -f b* 



X = 



A 2 m 2 — B 2 



and 



B 2 5 ± mABVB 2 — A 2 m 2 4- ¥ 



A 2 m 2 — B 2 
since Y = mX -j- b. 

The rational parts of these values, 

A?mb 



A 2 m 2 — B 2 




are the coordinates of M, the 
middle point of the chord PP. 
For a given value of m, the 
ratio of these co-ordinates is 
constant; for they satisfy the 

, . Y B 2 

relation — = - v or putting 

X mA 

m' for — ; Y = m'X. But 

mA 2 

this is the equation of a straight line passing through the centre, 
or of the diameter MC in the figure. Hence, a system of parallel 
chords of the hyperbola is bisected by a diameter. 



239. The condition of tangency is b 2 = A 2 



B 2 , because this 



value of b 2 makes the radical equal zero. When this condition is 



176 THE HYPERBOLA. 

fulfilled, the points P, P and M will coincide at P x the extremity 
of the diameter MC. Substituting the value b = ± l/A 2 m 2 _ B 2 , 
therefore, in the equation of the straight line, and in the co-ordi- 
nates of M, we have 

Y=wX± i/A 2 m 2 — B 2 

for the equation of a tangent line, and 

x _ - A 2 m y -B 2 



±l/A 2 m 2 — B 2 zhl/AW — B 2 

for the co-ordinates of its point of contact. 

The above radical value of b is imaginary when A 2 m 2 < B 2 j 
there are, therefore, directions in which no tangents can be drawn. 
But when A 2 m 2 ]> B 2 , two parallel tangents can be drawn, on£ 
touching each branch of the curve, as in the figure. 

If A 2 m 2 == B 2 , or m = ±: — , we find b = 0, and the equation 
A 
of the tangent becomes that of one of the asymptotes, X 2 and Y x be- 
coming infinite. Hence the asymptotes themselves fulfil the condi- 
tion of tangency. Since tangents are impossible when m is numerically 
less than these values, and possible when it is greater, the asymp- 
totes are the limits of the tangent as well as of the diameter ; and 
no tangent can be parallel to a diameter. 

240. Let the conjugate hyperbola be drawn. Since its equation 
is A 2 Y 2 — B 2 X 2 = A 2 B 2 , we can find the co-ordinates of its inter- 
sections with the secant line, by making the proper changes of sign 
in the equation solved in Art. 238. The term added to complete 
the square will be the same, and changing the sign of the term 
— A 2 B 2 (A 2 m 2 — B 2 ) in the numerator of the second member will 
have no other effect than that of changing the sign of the first two 
terms in the radical. Therefore the radical in the values of X and 
Y will become 

l/A 2 m 2 — B 2 -f b\ 

Since the rational parts are unchanged, we shall have the same co- 
ordinates for the middle point of the chord, and of course the same 
relation between them. Therefore parallel chords of conjugate 
hyperbolas are bisected by the same diameter. 



EQUATIONS OF THE TANGENT. 177 

The condition of tangency to the conjugate hyperbola, found by 
making the above radical equal zero, is b = -±- yB 2 — A 2 m 2 . 
Since the quantity under the radical sign is the negative of that 
in the last Article, tangents to this hyperbola are possible for those 
values of m which make the tangent to the other impossible. Thus, 
a tangent to either of these curves is parallel to a diameter of the 
other. 

241. Draw the diameter DD of the conjugate hyperbola parallel 
to the secant and tangents in the figure, its equation is Y == raX. 
The equation of the diameter T l P l was found to be Y = m'X, in 

B 2 

which m' = , therefore the direction ratios of these diameters 

raA 2 
are connected by the relation, 

. B 2 



A 2 

The form of this relation shows, as in the case of the ellipse, that 
each of the diameters Y = mX and Y = m'X bisects chords paral- 
lel to the other, and is parallel to tangents at the extremities or 
vertices of the other. They are called conjugate diameters of both 
of the hyperbolas. Since the axes are rectangular, m and m' are 
the tangents of the inclinations of conjugate diameters to the trans- 
verse axis. Their product being the square of the tangent of the 
inclination of either asymptote, each asymptote is conjugate to itself] 
or is the limit of a pair of conjugate diameters ; one, of those whose 
inclinations are both acute, the other, of those whose inclinations are 
both obtuse. In the equilateral hyperbola, B == A, therefore 
mm! = 1 ; in which case the acute angles are complements, the 
tangent of one being the cotangent of the other. 

Equations of the Tangent. 

242. We may express the equation of the tangent, Y = mX ± 
%/ AW — B 2 , in terms of the co-ordinates of its point of contact, 
by the same method that we used for the ellipse in Art. 216. Thus, 
referring to the values of X x and Y l5 in Art. 239, we see that the 
radical or value of b in the equation of the tangent (whether its 

B 2 B 2 X 

sign be positive or negative), equals j and that m === — — -. 

Y x A 2 Y X 

Making these substitutions and clearing of fractions, we obtain 

H* 



178 THE HYPERBOLA. 

A 2 YY X — B 2 XX X = — A 2 B 2 . 

To express the tangent at a given point in terms of a single 
arbitrary constant, instead of the two co-ordinates X x and Y 1? we 
use the auxiliary angle <j) corresponding to the point of contact. 
Then, by Art. 236, X l == A sec (p and Y x = B tan <p. Substituting 
and dividing by AB, we have 

A tan cp . Y — B sec <p . X = — AB, 

which represents a tangent whatever the value of (p. 

Examples. — Show that P x satisfies this last equation. 

Find by each equation the tangents at the vertices of the trans- 
verse axis, and at the extremity of the parameter. 

Prove that if an ordinate and a tangent, drawn from the same 
point of a circle or ellipse having A A' for an axis, meet the axis in 
R and T, and an ordinate to the hyperbola be erected at T ; then a 
tangent at the extremity of this ordinate will pass through R. ' 

243. The equation of the tangent to the conjugate hyperbola is 



the value of b being found by the condition of tangency in Art. 
240. Substituting this value of b in the expressions for the co- 
ordinates of M, Art. 238, which apply to each curve, we have 

X x = - and Y x 



± V B 2 — AW ± V B 2 — A'W 

Using these values of X x and Y 1} we obtain for the tangent, 

A ? YY X — B 2 XX X = A 2 B 2 , 

which differs from the formula found in the last Article only in 
the sign of the second member. 

To express this tangent in terms of the auxiliary angle <p, we 
must put Xi = A tan <p, Y x = B sec <p } for these are the co-ordi- 
nates of a point on the conjugate hyperbola, by Art. 236. Hence 

A sec <p . Y — B tan <p . X = AB 

is the equation of a tangent to A 2 Y 2 — B 2 X 2 = A 2 B 2 . 

Examples. — Prove that tangents to the conjugate hyperbola, at 
the points where it is cut by the tangent at the vertex A, pass 
through the other vertex A'. 



EQUATIONS OF THE TANGENT. 



179 



Find the tangents to each curve for the values (p = 90° and 
(p — 270°. (Multiply the equations by cos 0, before giving its 
value.) 

244. To express the equation of a tangent in the form x cos a -j- 
y sin a =p, we may use the same method as in Art. 193, for the 
ellipse. Or, since the inclination of the tangent line is 90° Hh a, its 



— cot a. 



Substituting in Y — ml 



V~& 



B 2 



and multiplying each member by sin a, we have (giving p the 
positive sign) 



X cos a -f- Y sin a = V A 2 cos 2 a — B 2 sin 2 a. 

Any tangent may be represented by this equation, if we give to a 
its proper value ; parallel tangents having values of a differing by 
180°, and therefore equal values of the perpendicular. Thus, 
a = 0° gives X == A, the tangent at the right vertex; a = 180° gives 
— X = A, the tangent at the left vertex. If A cos a = ± B sin a, 
we find zero for the perpendicular ; there is therefore a value of a 
in the first, and another in the second quadrant, for which the tan- 
gent passes through the centre. Between these values the tangent 
is impossible, since the perpendicular becomes imaginary. The 
opposite values of a in the third and fourth quadrants are also 
limiting values of a, between which the tangent is impossible. The 
limiting values may easily be shown to be the inclinations of per- 
pendiculars to the asymptotes. 

245. We may now demonstrate properties of the hyperbola, 
analogous to those of the ellipse in Art. 194. First : . 

The sum of the squares of perpendiculars from the centre upon 
perpendicular tangents is constant, and, equals A 2 — B 2 . 

For the inclination of CB 
being a, and that of CR', 
90° -f a, 

CR 2 = A 2 cos 2 a — B 2 sin 2 a 
and 

CR' 2 = A 2 sin 2 a — B 2 cos 2 a ; 
hence 
CR 2 -fCR' 2 = A 2 — B 2 . 




Since CRPR' is a rectangle, the square of the diagonal is equal 



180 THE HYPERBOLA. 

to the sum of the squares of two adjacent sides. Therefore CP 2 — 
A 2 — B 2 ; that is, the distance of P from the centre is constant, and 
its locus is a circle whose radius is |/A 2 — B 2 .* In the figure, 
A > B, and therefore CP is possible • but if A = B, CP = 0, and if 
A < B, CP is imaginary. Accordingly, to the rectangular hyper- 
bola there is but one pair of perpendicular tangents, namely, the 
asymptotes which intersect at the centre ; and to an obtuse hyper- 
bola there are no perpendicular tangents. 

The locus of the foot of a perpendicular from, a focus upon a tan- 
gent is the circle described on the transverse axis. 

For, in the figure, ED == c sin a. But CD 2 = CR 2 + ED 2 , 
hence 

CD 2 == A 2 cos 2 a -f (c 2 — B 2 ) sin 2 a = A 2 , or CD == A. 

The product of the perpendiculars from the foci upon a tangent 
is constant, and equals — B 2 . 

For the perpendiculars are c cos a — ]/A 2 cos 2 a — B 2 sin 2 a, and 
— c cos a — l/A 2 cos 2 a — B 2 sin 2 a (see Art. 73) ; and the pro- 
duct of these quantities is — B 2 . The negative sign of this product 
is due to the fact that the foci are always on opposite sides of the 
tangent. 

Conjugate Diameters. 

246. We found, in Art. 241, the relation 

, B 2 

mm = — 

A 2 

between the direction ratios of conjugate diameters ; that is, be- 
tween the tangents of the inclinations of diameters to conjugate 

* The values of CR 2 for the ellipse and for the hyperbola differ only 
in the sign of B 2 . If for B 2 in the former case, and — B 2 in the latter, we 
substitute A 2 — c 2 , we have for either curve, CR 2 = A 2 — c 2 sin 2 a. If 
another conic have the same foci, so that the value of c is the same, and if A / 
represents its transverse or major semi-axis, the value of CR /2 for a per- 
pendicular line, tangent to the second curve, will be CR /2 = A /2 — c 2 cos 2 a. 
The value of CP will still be constant, and the locus of the i ntersection of per- 
pendicular tangents will be the circle whose radius is V A 2 -\- A /2 — c 2 . If 
A / = c this second conic reduces to the line FF', and its tangents become 
lines passing through the foci, which accordingly intersect the tangents to 
the first conic on the circle whose radius is A. 



CONJUGATE DIAMETERS. 181 

hyperbolas, of which each is parallel to tangents at the vertices of 
the other. 

'. Let <p denote the auxiliary angle of any point on the hyperbola j 
then its co-ordinates are X = A sec 4', Y = B tan <p. The direc- 

Y 

tion ratio of the diameter passing through this point is m = — === 

-X. 

B tan <p __ _. _ , . i 

. J\ow the co-ordinates of a point on the conjugate hyper- 

A sec <p 

bola are X = A tan <p, Y = B sec <p ; therefore the direction ratio 

of a diameter to this curve is of the form . Supposing the 

A tan <[> 

value of <p to be the same for two points, one on each curve, the 
product of the direction ratios of diameters drawn to these points 
will be 

B tan ^ B sec <p _ # 

A sec <p A tan <p A 2 

therefore the diameters will be conjugate. Hence, the vertices of 
conjugate diameters correspond to the same value of <p. 

247. Let P and P' be the vertices of a pair of conjugate diame- 
ters, we can now express both co-ordinates of each point in terms 
of (p. Adding the squares of the co-ordinates to obtain the squares 
of the distances from the centre, we have 

CP 2 = A 2 sec 2 <p -f B 2 tan 2 <p 
and CP' 2 = A 2 tan 2 -f B 2 sec 2 <p. 

Subtracting, CP 2 — CP' 2 = A 2 — B 2 , 

since sec 2 — tan 2 = 1 j that is, the difference of the squares of con- 
jugate semi-diameters is constant, and equal to A 2 — B 2 . 

In the equilateral hyperbola, this difference is zero j therefore all 
its conjugate diameters are equal. When A > B, or the trans- 
verse axis is greater than the conjugate, every diameter is greater 
than its conjugate. 

248. If for B 2 , in the expression for CP' 2 , we put its value 
c 2 — A 2 , we derive CP' 2 = c 2 sec 2 <p — A 2 = e 2 x 2 — A 2 , putting Ae 
for c, and for A sec 0, x denoting the abscissa of the point P. But 
e?x 2 — A 2 is the product of the focal distances of P, r and / in Art. 
221. Hence, the product of the focal distances of the vertex of a 

16 



182 THE HYPERBOLA. 

diameter equals the square of the conjugate semi-diameter. This 
was also proved for the ellipse in Art. 197. 

In the equilateral hyperbola, the conjugate diameters being equal, 
CP 2 = r/, or the distance of a point on the equilateral hyperbola 
from the centre is a mean proportional between its distances from 
the foci. 

249. The equations of tangents in terms of 0, which we found 

in Arts. 242 and 243 — namely, 

A tan 4> . Y — B sec . X = — AB 
and A sec <p . Y — B tan <p . X == AB, 

are the equations of tangents at P and P' the vertices of conjugate 

diameters, since the value of <p 

for these points is the same. In 

the figure, P and P' are so 

taken that <p is in the first 

quadrant. 

These tangents intersect the 
asymptote AY = BX in the 
same point. For, combining 
AY = BX with the first equa- 
tion, eliminating successively Y and X, we have for the co-ordi- 
nates of intersection, 

A _ „ B 

X == and 




sec <p — tan <ft sec <p — tan <p 

Combining the equation of the asymptote with the second equation, 
we find the same values of X and Y ; that is, the same point of 
intersection. 

The four tangents at the vertices of the conjugate diameters 
evidently form a parallelogram with centre at C. Each side is equal 
to the diameter to which it is parallel, and is bisected at the point 
of contact. The above are the equations of two of the sides, and 
we have seen that they meet one of the asymptotes in the same 
point. The equations of the other sides of the parallelogram, which 
are parallel and equally distant from the origin on the other side, 
will be found by merely changing the sign of the second member. 
Now find the intersection of the other asymptote, AY = — BX, 



CONJUGATE DIAMETEKS. 183 

with the tangent at P, and also with the tangent parallel to that 
at P'. We find in both cases 

X = and Y = — 



sec <p -f- tan <p sec <p -j- tan <p 

that is, these tangents also meet one of the asymptotes in the same 
point. Hence the co-ordinates found in this Article are those of 
T and T', the angular points of the parallelogram, and the diagonals 
CT and CT' are the asymptotes. Therefore, the asymptotes are the 
diagonals of the parallelogram formed by tangents at the vertices 
of conjugate diameters. 

From this it is evident that, if a tangent cut the asymptotes, the 
intercepted portion is equal to the parallel diameter of the conju- 
gate hyperbola, and is bisected at the point of contact. 

250. The distances of the points T and T" from the centre, are 
the square roots of the sums of the squares of their co-ordinates. 
Hence we have for the semi-diagonals (since A 2 -|- B 2 = c 2 ), 

CT = and CT' = . 



sec <p — tan (p sec ip -j- tan <p 

Therefore CT X CT' = c 2 ; 

that is, the product of the intercepts made by a tangent upon the 
asymptotes is constant, and equals c 2 . 

It follows that the triangle TCT', formed by a tangent and the 
asymptotes, is constant in area; for one angle and the product of 
the including sides is constant. When the point of contact is at 
the principal vertex, CT and CT' are each equal to c, and the area 
of the triangle is AB, the product of the semi-axes (see Fig. Art. 
234). Therefore four times this triangle, or the parallelogram of 
conjugate diameters, equals the rectangle of the axes. 

Examples. — Find the values of the conjugate semi-diameters, 
corresponding to (p = 0°, 30°, 45° and 60°. 

Find the value of <p for which CP = c, also that for which 
CP' = c. (Give the values of tan <p in each case.) 

Find the equations of the diameters CP and CP' by means of the 
co-ordinates of P and P', and show from these equations that they 
are parallel respectively to the tangents at P' and P. 



184 THE HYPERBOLA. 

Show by the formula for the distance of two points, that 
TT' = 2CP. 

Prove that the transverse semi-axis is a mean proportional be- 
tween the abscissas of T and T'. 

Find the co-ordinates of T directly as the intersection of the 
tangents of Art. 249. 

Prove by the values in Arts. 247 and 250, that CT 2 + CT' 2 = 
2 (CP 2 + CP' 2 ). 

Tangent and Focal Lines. 

251. The equations of the lines joining a given point of the 
hyperbola to the foci may be expressed in terms of the auxiliary 
angle j and then one of the equations of lines bisecting their angles, 
found as in Art. 200 for the ellipse, would be that of a tangent line. 
It may, however, be proved that the tangent makes equal angles 
with the focal lines, in the following more simple manner. 

Let P X D be a tangent, and let r and r' denote the focal distances 
of the point of contact P 1# Then by Art. 221, 

r == eX x — A and r' = eX t -\- A. 

Draw the perpendiculars FD and 
FD' from the foci. The ratios 
of these perpendiculars to the 
corresponding focal distances are 
the sines of the angles at P x . To 
find them we must express the 
perpendiculars in terms of the 
co-ordinates of P v Therefore, using the equation 

A 2 YYl — B 2 XX X + A 2 B 2 =- 

for the tangent, and the method of Art. 73 for the perpendicular 
from any point, we have for the foci F, (Ae, 0) and F', ( — Ae, 0), 

gD = -Ag(eX 1 -A) ^ F3y= A B'(eX I + A)^ 
X/A 4 Y X 2 + B 4 X X 2 i/A 4 Y 1 2 + B 4 X x 2 

These values are of opposite signs because the foci are on opposite 
sides of the tangent. Dividing their numerical values respectively 
by r and /, we have the same value for sin FP X D and sinF'P x D' ; 




TANGENT AND FOCAL LINES. 185 

therefore the angles FPjD and F'PjD' are equal. A similar proof 
may be used in the case of the ellipse. 

252. It has been proved, in Art. 245. that the product of the 
perpendiculars FD and F'D' is numerically equal to B 2 , and in Art. 
248, that the product of the focal distances FP X and F'P X is equal 
to the square of the semi-diameter conjugate to CP X or parallel to 
the tangent. Denoting this semi-diameter by CP', as in previous 
Articles, we have 

sin FPJ) sin F'P^' = - , or sin FP X D = - , 

of 2 cpV 

since these angles are equal. This expression applies also to the 
ellipse, for the same properties were proved of the focal perpen- 
diculars and distances, in Arts. 194 and 197. Hence, both for the 
ellipse and for the hyperbola, this expression gives the sine of the 
angle made at any point of the curve by a tangent with either of 
the focal distances.* In both cases, the greatest value of the sine 
of this angle is at the vertices of the major or transverse axis, for 
which CP'=B, its least possible value; the corresponding value 
of the angle being 90°. In the ellipse the least value of the angle 
occurs at the vertices of the minor axis, where CP' = A, which is 
its greatest possible value; but in the hyperbola, since CP' may be 
increased without limit, the angle may be diminished without limit. 

253. The equation of the normal at a given point ; that is, the 
straight line perpendicular to the tangent at the point of contact, 



* The expression for the tangent of this angle in terms of the co-ordi- 
nates of the point is of a simple form. Thus, denoting the angle by r, 

"D T> 

sin r = , therefore tan r = . But V CP /2 — B 2 = c sin <p 

CP' /CP 72 — B 2 

for the ellipse, and = c tan ip for the hyperbola. (See values in Arts. 196 

T>2 

and 247.) Hence in each case, tan r = — , since y = B sin <p or B tan if>. 

cy 

This expression gives the acute angle at P when y is positive, and the ob- 
tuse value when it is negative. In polar co-ordinates at the focus we have, 

since y is a perpendicular to the axis, tan r = " , or substituting the 

err sin 6 

value of r, Art. 229, tan r == . 

e sin 

16* 



186 THE HYPERBOLA. 

may be found by the formula y — y l = m{x — x x ) for a line pass- 
ing through P x . To make this line perpendicular to the tangent, 
we must substitute for m the negative of the reciprocal of the value 

Wx 

of m in the equation of the tangent, which is -. Hence, we 

have y—y x = — — (x — x x ), or 

B 2 x,y = — A 2 y x x -f 6%^, 

which is the equation of a normal, when Pj is a point on the curve. 
Since the value of m in the equation of the tangent to the conju- 
gate hyperbola, Art. 243, is of the same form, the above is the 
equation of a normal to the conjugate hyperbola when P x is a point 
of that curve. It is easily shown that P x always satisfies the equa- 
tion of the normal.* 

Examples. — Show that the product of the values of FD and 
F'D' is — B 2 (using the condition that P x satisfies the equation of 
the hyperbola). 



* Since the normal is perpendicular to the tangent, it bisects, in the case of 
the hyperbola, the exterior angle of the focal lines. Hence, if an ellipse and 
an hyperbola have the same foci, the tangents to the ellipse at the inter- 
sections are normal to the hyperbola. 

Let A and B denote the semi-axes of the ellipse, and A r , W those of the 
hyperbola. Then A 2 y 1 y -f- B 2 x x x = A 2 B 2 and W 2 x x y -f- A. /2 y x x — c 2 x 1 y 1 , 
are equations of the same line, tangent to the ellipse and normal to the 
hyperbola at P a . Hence the ratios of the coefficients and absolute terms in 
these equations (that is, the values of the intercepts and direction ratio as 
determined by them) are equal. Equating the values of x , and those of 

AA' BB' 

y , we obtain x x = ± and y x = db , for the co-ordinates of in- 

c c 

tersection. Equating the direction ratios (supposing x x and y x positive), 

y, BB' t> x Vi B sin 6 B / tan V ; d i -. ,- ^ i 

— = . But *i = = -. Substituting these values 

x i A A 7 x i A cos $ A / sec V 

successively, we find tan <f> = — , and sin V = — > Hence, the eccentric 
A / A 

angle of the point in which an ellipse is cut by a confocal hyperbola is the 

inclination of the asymptotes of the hyperbola; and the auxiliary angle in 

the hyperbola corresponding to the same point is the inclination of a line 

joining the focus with the vertex of the minor axis of the ellipse. 



HYPERBOLA REFERRED TO CONJUGATE DIAMETERS. 187 

Prove that the parts of the normal intercepted between the axes, 
between P x and the axis of X and between P x and the axis of Y 
have the same ratios as c 2 , B 2 and A 2 . (This property belongs also 
to the normal to the ellipse.) 

Find the values of A 2 and B 2 for the conic section to which 

A 2 v 
y — y x = m(x — Xy) is normal at P v (Since m = — in the 

B 2 

normal, — may be expressed in terms of m, x x and y x \ and then the 

A 2 

value of A 2 may be found by an equation of condition expressing 

B 2 

that the curve passes through P x . If — as thus found would be 

A 2 
negative, we must use the normal of Art. 201, and determine an 
ellipse.) * 

Determine in this way the conic normal to y = 2x -f- 2 at (1, 4) 
and verify the result by finding the normal at the given point. (This 
curve being an ellipse in which A 2 < B 2 , weniust give the negative 
sign to c 2 , which was put for A 2 — B 2 in the equation of the nor- 
mal, Art. 201.) 

Determine the conic normal to y -j- 2x = 5 at (2, 1). (The value 
found for A 2 will be zero, therefore the equation reduces to that of 

B 2 

the asymptotes determined by the value found for — .) 



Hyperbola Referred to Conjugate Diameters. 

254. The equation of the hyperbola as referred to a pair of con- 
jugate diameters may be found by the method that we used for the 
ellipse in Art. 202. Thus, adding the squares of the co-ordinates 
of M, or rational parts of the values of X and Y in Art. 238, for 
the square of the oblique abscissa CM ; and adding the squares of 
the radical parts for PM 2 , we have 

_ 5 2 (A 4 m 2 + B 4 ) _ (1 -f m 2 ) A 2 B 2 (B 2 — A 2 m 2 -f b 2 ) 

~ ( AW — B 2 ) 2 ' ( A 2 m 2 — B 2 ) 2 

In these equations m is constant, and we have to eliminate b 2 . To 
simplify the result we introduce as new constants the values of the 
conjugate semi-diameters. Let A denote that which is measured 



188 THE HYPERBOLA. 

on the new axis of X, which we will suppose to cut the hyper- 
bola, as in the figure. Then 

A 2 m 2 — B 2 is a positive quan- NX. / /(/ 

tity, and if we give this value \N. / ^/J? 

to b 2 , the line PP will become V\ / /W/ / m/J--x 

a tangent as explained in Art. \ •^^^^^Zl!- 

239, and accordingly the value ■^~^~^//' ^^L \J 

of Y 2 becomes zero. The cor- // / D '^ S C\. 

responding value of A" 2 is J. 2 ; / ^xX 
therefore 

A 2 m 2 — B 2 ' 

Since the curve does not intersect the new axis of Y, b = 0, 
which makes A 2 = 0, will be found to make Y 2 negative. Let B 
denote the conjugate semi-diameter, or intercept of the conjugate 
hyperbola on the axis of Y. Now in Art. 240, we found that the 
values of X and Y for the conjugate hyperbola differ from those in 
Art. 238 only in the radical, which becomes l/AV* 2 — B 2 -f ° 2 - 
Hence, for the conjugate hyperbola, the value of A 2 is the same as 
that above, but 

_ (i j- m *)A 2 B 2 ( A 2 m 2 — B 2 -f 6 2 ) 
~~ (A 2 m 2 — B 2 ) 2 

From this we derive, by making b = 0, 

^_ q + ^ 2 )A 2 B 2 

A 2 m 2 — B 2 

Dividing X 2 by A 2 , and the first value of Y 2 by B 2 , we derive 

Y 2 B 2 — A 2 m 2 -f b 2 

and — == — • 

£* A 2 m 2 — B 2 

or A 2 Y 2 — B 2 X 2 == — A 2 B\ 

an equation of the same form as the original rectangular equation. 
Dividing the second value of Y 2 by B 2 , we obtain in the same 
way for the equation of the conjugate hyperbola, 



X 2 


b 2 




A 2 ~ 


A 2 m 2 — 


B 2 


Hence — 

A 2 


F2 _ 
B 2 ~ 


= 1 



HYPERBOLA REFERRED TO CONJUGATE DIAMETERS. 189 



— = 1 or A 2 Y 2 — B 2 X 2 = A 2 B 2 

B 2 A 2 



which is also of the same form as the rectangular equation of the 
same curve. 

255, If the value of A 2 m 2 — B 2 had been negative, the new axis 
of X would have cut the conjugate hyperbola, and that of Y would 
have cut the curve of the figure. The values of A 2 and B 2 would 
be in form the negatives of the values of the last Article; that of 
A 2 being found by making b 2 = B 2 — A 2 m 2 , which makes the line 
PP tangent to the conjugate hyperbola, and that of B 2 being found 
by means of the first value of Y 2 . Hence the equation of the 
hyperbola in the figure would take the second of the above forms, 
and that of its conjugate would take the first form. 

Therefore, dropping the distinction between oblique and rectan- 
gular co-ordinates, we may regard A 2 Y 2 — B 2 X 2 = — A 2 B 2 and 
A 2 Y 2 — B 2 X 2 =A 2 B 2 as the equations of two conjugate hyper- 
bolas, of which the axes are a pair of conjugate diameters. Since 
conjugate diameters of a given hyperbola may be found making any 
given angle with each other, either of these equations may be made 
to represent an hyperbola of any shape, by giving proper values to 
A and B, whatever be the inclination of the co-ordinate axes.* 

256. If on the tangent at the vertex A we lay off AD and AD', 
each equal to the conjugate semi-diameter B, and join CD, CD', 
the oblique equations of these lines will be 

AY=BX and AY= — BX, 

for they pass through the origin and the points (A, B) and (J., — B) 
respectively. But these lines are evidently the diagonals of the 
parallelogram constructed, as in Art. 249, on a pair of conjugate 
diameters, hence they are the asymptotes. These equations are 
also of the same form as the rectangular equations of the same lines, 
and therefore the compound equation 

* This is not true of the corresponding equation of the ellipse, since 
there is a limit to the obliquity of its conjugate diameters. See Note to 
Art. 206. The general central equation Ax 2 -f- Oy 2 -f- F — cannot re- 
present a conic having an eccentricity less than that of the ellipse making 
equal intercepts on the axes. 



190 THE HYPERBOLA. 

A 2 Y 2 — B 2 X 2 =0 

represents both asymptotes of the curves A 2 Y 2 — B 2 X 2 = =f- A 2 B 2 , 
whether the axes are rectangular or oblique. When A = B, the 
equations of the asymptotes reduce to Y = X and Y = — X, which 
are in all cases the equations of lines bisecting the angles between 
the co-ordinate axes. Therefore when A = B, the asymptotes are 
at right angles and the hyperbola is rectangular. But when A and 
B are unequal, the asymptotes are oblique to each other ; and the 
hyperbola is acute when the transverse semi-diameter (that which 
is intercepted by the curve) is greater than its conjugate; for by 
Art. 247, the transverse semi-axis will then be greater than the 
conjugate. 

257. When the hyperbola and its conjugate are both drawn, or 
when the equation is given, so that the values of A and B are 
known, the asymptotes may be drawn, and then the axes of the 
curve may be drawn bisecting their angles. But when the curve 
and its centre are given, the axes may be constructed geometrically 
by the method pointed out for the ellipse in Art. 203 ; for the 
property of supplementary chords there proved evidently extends to 
the hyperbola also. If the centre is not given, it may be found 
geometrically, by drawing a diameter bisecting parallel chords, and 
then bisecting the diameter, or if one branch only of the curve is 
given, by finding the intersection of two diameters. 

258. Hyperbolas having their axes proportional have the same 
eccentricity, and are said to be similar. As in the case of similar 
ellipses, the value of r 2 , Art. 226, shows that if their transverse 
axes are parallel, all the parallel diameters will have the same 
ratio. 

Since the inclination of the asymptotes determines the ratio of 
the axes, hyperbolas having the same transverse axis and the same 
asymptotes are similar, and their conjugate hyperbolas are also 
similar. 

Let n = — , then the central equation of any hyperbola may be 
B 

put in one of the forms, 

X 2 — n 2 Y 2 = ± A 2 , 
in which n determines the asymptotes/, which are the lines X = nY 



AXES PARALLEL TO CONJUGATE DIAMETERS. 191 

and X = — nY. Equations of this form, in which the value of n 
is the same, therefore represent hyperbolas having the same asymp- 
totes. If the sign of the second member is the same in each, they 
are similar, being situated in the same angles. But if the second 
members have opposite signs, they are situated in the supplemental 
angles of the asymptotes ; one being acute and the other obtuse, 
except when n=l, in which case both are rectangular. 

If the value of n is fixed, while that of the second member varies, 
the asymptotes will be fixed. Then, as the second member is 
diminished, the hyperbola will approach nearer and nearer to 
the asymptote. When it becomes zero, we have the equation 
X 2 — n 2 Y 2 = 0, equivalent to X = ± nY, the equations of the 
two asymptotes. Therefore the hyperbola is said to vanish into a 
pair of straight lines. 

Axes Parallel to Conjugate Diameters. 

259. The equation of an hyperbola whose centre is at P' is found 
by substituting for X and Y, in the central equation, their equiva- 
lents x — x' and y — y' . Hence 

A 2 (y — y' y — B 2 (x — xj = =F A 2 B 2 , 
and (x — x'f — n 2 (y — y') 2 == ± A 2 , 

are the equations of an hyperbola having a pair of conjugate diame- 
ters parallel to the co-ordinate axes. The first is expressed in terms 
of the lengths of the semi-diameters, the second in terms of a ratio 
?i, which determines the direction of the asymptotes, and a quantity 
which determines the size of the curve, or its distance from its 
asymptotes. If we put zero in place of the second member in 
either of these equations, we produce the equation of the asymp- 
totes. Expanding the last equation, we have 

x* _ n y _ 2x'x + 2n 2 y'y + x' 2 — n 2 y' 2 q= A 2 = 0.- 

From this we see that the general equation 

Ax 2 + Cy 2 + T>x -f- Ey -f F = 

represents an hyperbola, when A and C, the coefficients of x 2 and y 2 , 
have opposite signs. 



192 THE HYPERBOLA. 

260. This is the general equation of the conic section having a 
pair of conjugate diameters parallel to the co-ordinate axes, includ- 
ing, as shown in Art. 209, the parabola having its axis parallel to 
either of the co-ordinate axes. A conic fulfilling this condition 
may be found, which shall pass through four given points ; for the 
ratios of the five coefficients may be determined by four equations 
of condition. Thus, we may assume the equation in the form 
x 2 -f- cy 2 -f- dx -\- ey -\-f= ; and then by substituting for x and 
y the co-ordinates of each of the given points successively, we shall 
have four equations by which to determine c, d, e and/ It may 
happen that these equations will be found incompatible ; this will 
occur when the conic is a parabola of the form Cy 2 -\- Dx -f- Ey -\- 
F = 0, in which case the proper coefficient of x 2 being zero, the 
equation cannot be expressed in the assumed form. If the four 
points are in one straight line, we shall find A = and C = 0. 

261. To determine the centre and semi-diameters of the curve 
from the values of c, d, e and /, we may use the equations of Art. 
208, prepared for the ellipse. The only difference is, that c 
(being negative) is the value of — n 2 , in the hyperbola, and that 

\ld 2 -\- - ) — f may be the value of either ± A 2 . Therefore the 

equation will represent an hyperbola, whether the value of this 
quantity be positive or negative ; but if it be zero, the equation 
will represent a pair of straight lines. When this quantity equals 
zero in the equation of an ellipse, the curve vanishes into a 
single point, as shown in Art. 210. To find the condition that 
an equation of the above general form shall represent a pair of 

****** appoint, substitute in^ + ^O, 

D E 

the ratios — for d, — for e, etc. The result is 

A A 

AE 2 + CD 2 — 4ACF = 0. 

If we suppose C to be positive, the sign of the expression in the 
first member has not been changed by clearing of fractions, which 
was effected by multiplying by 4A 2 C. Therefore making C posi- 
tive in a given equation, this expression will he positive for a real 
ellipse or an hyperbola in the first form ; that is, one which cuts the 



AXES PARALLEL TO CONJUGATE DIAMETERS. 193 

diameter parallel to the axis of X. If the expression is negative, 
the ellipse is imaginary, or the hyperbola is of the second form ; that 
is, the equation may be reduced to one of the forms (x — x') 2 ft 
n\y-y') 2 = -A 2 . 

Examples. — Determine the form of x 2 — 2y 2 — 4x -j- 4y = 0; 
f x 2_2y ! — 4x -f 4y -f 10 = 0; of x 2 + 2/ -f 4x — 4y -f 
6 = 0. 

262. If A = 0, the equation generally represents a parabola j but 
wc found in Art. 156 that if D — it represents a pair of straight 
lines. Now A = and D = will satisfy the condition of the 
last Article independently of the value of F. In like manner, 
C = and E = 0, which makes the equation represent lines paral- 
lel to the axis of Y, satisfies the condition. It is also satisfied by 
A = and C === 0, which reduces the equation to the first degree. 
Hence the above is the general condition that Ax 2 -f- Cy 2 -f- Dx -j- 
Ey -(- F = shall represent straight lines. 

If in an equation representing an hyperbola we suppose the 
value of F to vary, the co-ordinates of the centre and the direction 
of the asymptotes will not be changed. Therefore, if in such an 
equation we give to F a value determined by the condition for straight 
lines, we shall obtain the equation of the asymptotes. Thus, given 
the hyperbola x 2 — 2y 2 — 4x -f- 4y -f- 10 = 0, we substitute the 
values of all the coefficients but F in the condition. This gives 
16 _ 32 -f 8F == 0, hence F == 2, and x 2 — 2y 2 — 4x -j- 4y -f 
2 = is the equation required. 

263. The co-ordinates of the centre, in Art. 208, become by sub- 
stituting their general values for c, d and e, 

D A / E 

x = and v = • 

2A * 2C 

From these values it is evident that if D = the centre is on the 
axis of Y, and if E = it is on the axis of X. If both D = 
and E = 0, it is at the origin. But if A = 0, the value of x' be- 
comes infinite; and if == 0, the value of i/ becomes infinite. 
Therefore the centre of a parabola is said to be at an infinite dis- 
tance. If A = and D = 0, x' takes the indeterminate form ; 

E 

therefore any point which satisfies y = — — , that is, any point on 

20 
the line 2Cy -f- E = 0, may be regarded as the centre of Qy 2 -j- 
17 I 



194 THE HYPERBOLA. 

Ey -j- F = 0, which represents parallel lines. The diameters of 
the parabola are all parallel or intersect at infinity ; but for parallel 
lines, they are all coincident. 

The centre of a pair of intersecting straight lines is their point 
of intersection, and that of an infinitesimal ellipse is the single 
point which satisfies the equation. Therefore the condition that 
the equation shall be satisfied by the centre is equivalent to the 
condition of Art. 261, as will be found by substituting the values 
of x' and y f for x and y in the general equation. 

264. The equation of the asymptotes of the hyperbolas of Art. 
259, is (x — x'y — n 2 (y — y') 2 = 0. Hence they are the two 



lines X; — oi "=s= d= n \y — ?/), in which n = + If we let 

-A 

these equations may also be written in the form 



v= 



y — y = dz ^ (x — x'). By this value of m, and the values of 
x' and y, in the last Article, we may form the equations of the 
asymptotes separately. Thus, given x 2 — 2y 2 — 4x -j- 4y -j- 10 = 0, 
we find x' = 2, y' = 1, m — -j/i, hence the asymptotes are 
y — 1 ==■ ± ]/J (x — 2). The compound equation formed from 
these by the method of Art. 81, will be found identical with that 
found for the same curve in Art. 262. 

Since for an ellipse A and C have the same sign, the above value 
of m will be imaginary ; therefore the ellipse is said to have imag- 
inary asymptotes. The compound equation will then be satisfied 
by a single point, namely the centre. We have hitherto called this 
the equation of an infinitesimal ellipse, because it is the vanishing 
case of an ellipse of given shape ; we may now consider it the equa- 
tion of two imaginary straight lines passing through the centre, 
whose imaginary direction ratios determine the direction of the 
axes and the shape of the ellipse. 

For the parabola in which A = 0, we find the direction ratio 
m = ± 0, which shows that the directions of both asymptotes 
coincide with that of the axis of X, but since the parabola has no 
centre, no asymptotes can be found. The directions of asymptotes 
are those of straight lines which cut the conic in only one point. 
For the hyperbola there are two such directions, for the parabola 
but one, and none at all for the ellipse. 



results of transformation. 195 

Results of Transformation. 

265. If the equation Ax 2 -f Cy 2 -f Vx -f % -f F == be trans- 
formed by changing the direction of the axes, the new equation 
will in general contain a term involving xy, the product of the 
variables. Since the formulae of Case IV., Art. 90, give values of 
x and y containing no absolute term, it is evident that the coeffi- 
cients A and C will appear only in the terms of the second degree 
of the transformed equation. Now the shape of the curve and 
directions of the asymptotes and axes depend only upon A and C ; 
they must, therefore, in every equation representing a conic section, 
depend only upon the terms of the second degree. We cannot, 
however, by these terms alone ascertain whether the equation re- 
presents a pair of real or of imaginary straight lines, or distinguish 
between the equations of real and imaginary ellipses. In the next 
Chapter we shall discuss the general equation of the conic section 
containing the term xy; the fact that that term is wanting in the 
present equation expresses the condition that the diameters parallel 
to the co-ordinate axes are conjugate. 

266. Change of origin without change in the direction of the 
axes does not affect the terms of the second degree, upon which 
we may consider the direction and shape of the curve to depend. 
But it does affect the terms of the first degree, upon which the co- 
ordinates of the centre, or the two elements of the position of the 
curve, depend. 

The absolute term of the equation is not altered by changing the 
direction of the axes ; but when the origin is moved to P', the new 
value of F, which we denote by F', is found to be 

F = Ax' 2 + Cy' 2 -f Vx' If E/ -f F, 

that is, the result of substituting the co-ordinates of the new origin 
in the first member of the given equation. (See Art. 100.) 

This quantity becomes zero for a point on the curve, in which 
case we know that the absolute term should be zero. 

267. Supposing the axes to be rectangular, and transforming to 
polar co-ordinates, we have 

(A cos 2 -f C sin 2 0) r 2 -f (D cos -f E sin 0) r -f F = 0, 



196 THE HYPERBOLA. 

in which the initial line is parallel to one or other of the principal 
axes of the curve. This equation gives in general two values of r 
corresponding to any value of 6. Now the product of the roots of 
a quadratic equation is the absolute term divided by the coefficient 
of the highest power. Therefore the product of the two values of 
r or segments of a chord passing through the pole is 

F 




A cos' 6 -f C sin 2 6>' 

where 6 is the inclination of the chord to the axis of X. Since, by 
the last Article, A and C are unchanged by transformation to a new 
origin, the products of the segments of parallel chords passing 
through different points are proportional to the values of F for 
those points ; that is, to the results obtained by substituting the co- 
ordinates of the points in the first member of the equation. 

268. The above expression is the value of PA X FB, if F is the 
absolute term corresponding to the 
point P in the figure. In gene- 
ral this is a varying quantity. 
But when A = C (which makes 
the curve a circle, since the axes 
are rectangular), it becomes con- 
stant, as shown in Art. 128. If 
P be taken at the focus, PA 
and PB will be the two values 

of r in Art. 229, whose product " is proportional to the 

1 — e 2 cos 2 

focal chord, Art. 232. Therefore the products of the segments of 
chords passing through the same point in different directions are 
proportional to the parallel focal chords. 

If A and C have the same sign, it is evident that the expression 
for PA X PB cannot be made to change sign by varying the value 
of 0, being always negative for a point within the ellipse, because the 
segments are measured in opposite directions from P, and positive 
for a point without the curve, because PA and PB are measured in 
the same direction. But when the curve is an hyperbola, the pro- 
duct will change sign, passing through the value infinity, when the 
line is parallel to an asymptote, as at PC. Thus, for a point without 






INTERSECTIONS OF CONICS. 197 

the curve (that is. a point from which a tangent may he drawn, as 
P, in the figure), it has the same sign as the parallel focal chord 
which we found, in Art. 232, to be positive when it cuts a single 
branch, and negative when it cuts both branches, or is parallel to a 
diameter of the curve. For a point within either branch of the 
curve, the reverse would take place, the value of F changing sign 
as the point passes across the curve. 

269. Since the value of F, for a given point, may be computed 
by the formula of Art. 266, we have the means of finding the value 
of this product for a given point and given value of d, without 
performing the transformation, when the axes are rectangular in the 
given equation. The same method will of course give the square 
of a tangent, as PD in the figure. 

The combined equation of Art. 213 may now be regarded as ex- 
pressing that the value of PA X PB for one of the given ellipses, 
equals - — k times its value for the other, for the same value of 6. 
For the expressions combined are the values of F for the point P, 
and the values of A and C in the denominators are the same. The 
straight line (d — d') x -f- (e — e')y -\-f — /' = is therefore the 
locus of the point P, when these products are equal, and it bisects 
the common tangents to the two conies. 

Intersections of Conics. 

270. If we eliminate one of the variables between two equations 
of the second degree, the result will be an equation of the fourth 
degree, containing the other variable. Although such an equation 
cannot be solved by the rules of Common Algebra, we know that in 
some cases it can be satisfied by four real values of the unknown 
quantity. Therefore the loci of two equations of the form 

Ax 2 -J- Cf + Vx -f Ey -f- F = 

may intersect in four points. We shall use the term conic to de- 
note the locus of this equation, which as we have seen may be a 
curve or a pair of straight lines. Let S represent the polynomial 
which constitutes the first member of the equation j then S = is 
the equation of a conic. Let S' = represent another equation of 
the same form, then the combined equation, 

s + m = o, 

17* 



198 THE HYPERBOLA. 

will be an equation of the same form ; that is, it will consist of 
terms containing the squares and first powers of the variables, and 
an absolute term. But it is satisfied by those values of x and y 
which satisfy both S = and S' = ; therefore it represents a 
conic passing through all the intersections of the conies S = 
and S' = 0. 

271. From the form of the equations combined, we know that 
the given conies have each a pair of conjugate diameters parallel 
to the co-ordinate axes. Since S -j- kS f = is an equation of the 
same form, the conic it represents fulfils the same condition, what- 
ever be the value of k. Therefore, if S = and S' — intersect 
in four points, S -j- &S' — represents a conic of which the direc- 
tion of a pair of conjugate diameters is fixed, and which passes 
through four fixed points ; k remaining arbitrary, so that another 
condition may be fulfilled.* For instance, we may determine k so 
as to make the term containing x 2 disappear, which will make the 
conic a parabola of the form Cy 2 -j- Dx -j- E y -f- F = ; that is, one 
whose axis is parallel to the axis of X. Or we can make the term 
containing y 2 disappear, and thus determine a parabola with axis 
parallel to the axis of Y. 

We may also determine k by the condition that the coefficients 
of x 2 and y 2 in S -f- k$' = shall be equal. This will make the 
conic an ellipse, in which the diameters parallel to the co-ordinate 
axes shall be the equal conjugate pair, but will not make it a circle 
unless the axes are rectangular. Again, we can make these coeffi- 

* It would seem from this that a conic of this form, which already fulfils 
one condition, might be found passing through jive points, whereas we saw 
in Art. 260, that it can only be made to pass through four given points. The 
explanation is that the four /zed points are not four given points ; and that 
passing through them really constitutes but three conditions, because any 
three of them imply the fourth. For let A, B and C be three given points 
and S=0, S / = the equations of two conies of the above form passing 
through them. Then S-J-&S' — represents all the conies of this form, 
which pass through A, B and C; because, by Art. 260, there is generally 
but one passing through these three and a given fourth point, and if we 
properly determine k, S + kS' = wil 1 become its equation. But S = and 
S / == evidently intersect in a certain fourth point, and S + kS' = 
always passes through that point. Hence also, four points and the condi- 
tion implied in the form of the equation do not always determine a conic. 



INTERSECTIONS OF CONICS. 199 

cients equal but of opposite signs, which will always determine an 
equilateral hyperbola. See Art. 256. 

272. If A, G, D, E and F are the coefficients in S = 0, and 
A', C, D\ E' and F', in S' == 0, then A + kA', C + kC, etc. 
will be the corresponding coefficients in S -f- &S' = 0. Therefore 
the conditions determining k, in the last Article, give rise to equa- 
tions of the first degree. But if we substitute these coefficients in 
the condition of Art. 261, the result is an equation of the third 
degree, to determine k so that the equation S -J- k$ f = shall 
represent either two straight lines or a single point. Now an equa- 
tion of the third degree may be satisfied by three values of k ; 
accordingly, when S = and S' = meet in four points A, B, C 
and D, the pairs of straight lines, AB and CD, AC and BD, AD 
and BC, constitute three cases in which the conic S -j- kS' — 
satisfies the condition. 

If S = and S' = intersect in only two points, the equation 
of the third degree will be satisfied by only one value of k; and 
that value will make S -f- k& = a pair of straight lines, one 
of which passes through the two points of intersection (or is a com- 
mon chord), and the other fails to meet either of the given conies. 
But if its equation were combined with each of the given equations, 
there would result the same imaginary values of x and y ; therefore 
the conies are said in this case to meet in two real and two imagin- 
ary points. If the conies merely touch each other in a single point, 
the common chord becomes a common tangent, and the two real 
points become coincident. If the conies do not meet at all, the 
straight lines are said to meet the curves in four imaginary points* 

273. When A : C : : A' : C' \ that is, when the coefficients 
of x 2 and y 2 , in S = and S' = 0, are proportional ; the value 

A 

k = — will reduce S -j- k%' = to an equation of the 

A U 

first degree. f But for other values of k it will represent a series 

* In this case, as well as when the intersection is in four real points, the 
cubic equation would have three real roots ; but two of the roots would now 
give infinitesimal ellipses or vanishing points. 

f This value of A; will of course satisfy the cubic equation, but one of the 
pair of lines is the line at infinity, which intersects hyperbolas with parallel 
asymptotes in two real points at infinity— namely, the intersections of the 



200 THE HYPERBOLA. 

of similar and parallel ellipses, like the equation of Art. 213, or a 
series of hyperbolas with parallel asymptotes, as that equation 
would if the negative sign were given to ri\ In the latter case, it 
may be proved, as in Art. 214 for the ellipse, that the common 
chord {d — d f ) x -j- (e — e'~)y -\- f — /' — is parallel to the 
diameters conjugate to the common diameter, or line joining the 
centres of the given conies. Now if the given hyperbolas have a 
common asymptote, this line must be parallel to it ; for the line 
joining the centres will be the asymptote, and an asymptote is con- 
jugate to itself. But a line parallel to an asymptote cuts an hyper- 
bola in only one point ; therefore, in this case, the hyperbolas will 
cut in a single point, and S -(- &S' = will represent a series of 
hyperbolas with a common asymptote.* 

274. Let the co-ordinate axes be rectangular ; then S -f- &S' = 
represents a series of conies whose axes are parallel ; for the axes 
are the only rectangular pair of conjugate diameters. Let S = 
and S' = intersect in the four points A, B, C and D. Now 
when S -f- &S' == becomes the pair of straight lines AB and CD, 
its axes become the lines bisecting the 
angles between these lines. Hence, 
the lines which bisect the angles be- 
tween the chords AB and CD are 
parallel to the axes of S = 0, or 
these chords are equally inclined in 
opposite directions to an axis of the 
conic. For the same reason, AC and 
BD are equally inclined to the same 
axis. Hence, if two intersecting 
chords, AB and CD, be drawn, making equal angles with an axis 

parallel asymptotes. See Art. 43. But it intersects a parabola in two 
coincident points, and an ellipse in two imaginary points, because the direc- 
tions of the asymptotes become, in these cases, respectively coincident and 
imaginary, as shown in Art. 264. If the conies are concentric also, both 
of the straight lines and all four of the intersections are at infinity. Com- 
pare Art. 133 and Note. 

* Since an asymptote is a tangent at infinity, these hyperbolas are said to 
have three points at infinity, two of which are coincident. Concentric conies 
have two pairs of coincident points at infinity. 





RECIPROCAL POLARS. 201 

of a conic, the chords AC and BD will also be equally inclined to the 
axis. It follows that the angles ACD and ABD will be equal when 
the chords are thus drawn. Since the conic may be a circle, of 
which any diameter may be taken as an axis, we have, as a special 
case of this theorem, the well known property of the equality of all 
the angles in the same segment of a circle. 

Reciprocal Polars. 

275. The form of the equation of the hyperbola, as referred to 
any pair of conjugate diameters, being the same as that of the rec- 
tangular equation used in Art. 237, the condition of tangency and 
equation of the tangent are of the same form in this case as when the 
curve is referred to its axes. Hence the equations of Arts. 242 and 
243, for a tangent in terms of the co-ordinates of its point of con- 
tact, apply when the axes are oblique. 

In general, the equations 

A 2 YYi — B 2 XX X = hF A 2 B 2 

represent straight lines which are called polars of P 1} with respect 
to the hyperbolas A 2 Y 2 — B 2 X 2 = =£ A 2 B 2 . It may be shown, as in 
Art. 217 for the ellipse, that if P 2 is on the polar of P l3 then P x is on 
the polar of P 2 ; that is, P x and P 2 are reciprocally polar. Again 
it may be shown, as in Art. 218, that the polar of a point on any 
diameter is parallel to the conjugate diameter, and cuts it at a dis- 
tance from the centre which forms a third proportional to the dis- 
tance of the point and the semi-diameter. 

The more general equation of Art. 219 evidently applies to the 
hyperbola as well as to the ellipse, and in general, 

Axx x + Cyy x + JD (x + xj + *E (y + y t ) + F = 

is the formula for the polar of P l5 with respect to the conic 

Ax 2 4- Qy 2 -f J)x -f Ey -f F = 0. 

276. Since the formula for the polar contains two arbitrary con- 
stants, it may in general be made the equation of any given line, 
by giving proper values to x x and y x . The point of which a given 
line is the polar is called its pole. To find the pole of a given line, 
that is, the proper values of x x and y x . select two points of the line, 

I* 



202 



THE HYPERBOLA. 



and find the intersection of their polars. If we denote the two 
points by P 2 and P 3 , the algebraic process will evidently be the 
same as that of finding P x by two equations of condition, expressing 
that its polar passes through P 2 and P 3 . This is a consequence of 
the reciprocal property above referred to. 

277. This reciprocal property, and the fact that the polar of a 
point on the curve is a tangent at that point, furnish the following 
geometrical definitions of the polars of given points and the poles 
of given lines. 

If tangents be drawn at the points in which a secant line cuts the 
curve, their intersection is the pole of the secant. For these tangents 
are the polars of two points on the secant. Thus P l3 in the figure, 
is the pole of P 2 P 3 . Every secant not passing through the centre 
has a pole, but a diameter has no pole, because the tangents at its 
extremities are parallel. 

If tangents be drawn through a given point without the curve, the 
line joining the points of contact is the polar of the given point. 
For these points are the poles of two lines passing through the 
given point. If the curve is an hyperbola and the given point the 
centre, the tangents become asymptotes, and the points of contact 
are at an infinite distance. 

Now if the polar of V x be constructed for any conic, it will con- 
tain the poles of all lines passing 
through P l7 for such points are re- 
ciprocally polar to P x . Hence we de- 
rive the following general property of p ^ 
the conic, sections : 

If tangents and secants be drawn 
through any point, the intersections of 
pairs of tangents , drawn at the points 
where each secant cuts the curve, will 
lie in a straight line with the first 
points of tangency. 

Thus, the polar of P x is the locus of 
all points which can be constructed in the same manner as P 2 and 
P 3 in the figure. If the point P x be taken within the conic, so 
that tangents cannot be drawn through it, its polar may be con- 
structed by joining the poles of any two lines drawn through it. 




HYPERBOLA REFERRED TO ITS ASYMPTOTES. 203 

Therefore, the locus of the intersection of tangents at the extremities 
of a chord drawn through a given point is a straight line, and this 
straight line is the polar of the given point. 

If the chord or secant drawn through P x is a diameter, the tan- 
gents are parallel to each other, and therefore parallel to the polar. 
From this it is evident that the polar is parallel to the diameter 
conjugate to that passing through the pole. 

Hyperbola Referred to its Asymptotes. 

278. It is proved in Art. 249 that the portion of the tangent in- 
tercepted between the asymptotes is bisected at the point of contact, 
and in Art. 250, that the product of the intercepts on the asymp- 
totes equals c 2 . These properties enable us to find the equation of 
the hyperbola as referred to its asymptotes. For let P be any point 
of an hyperbola whose asymptotes are the co-ordinate axes CX 
and CY. Draw the tangent 
at P, cutting the axes in T 
and T', and draw the ordinate 
of P. Then since PT' = *TT' 
by similar triangles y = ?CT 
and x = £GT'. But CT X 
CT' = c 2 ; therefore 

xy — ic 2 , 

or the product of the co-or- 
dinates equals one-fourth of the square of the distance from the 
centre to the focus. The form of this equation shows that the 
curve does not cut either of the axes, for neither x nor y can be 
made equal zero. Since the product of the co-ordinates is positive, 
the equation is also satisfied by points of which the co-ordinates 
are both negative, which constitute the other branch of the curve. 

279. The equation of the tangent TT' is most readily expressed 
in terms of the co-ordinates of its point of contact. For this pur- 
pose denote the co-ordinates of P, in the figure, by x x and y v Then 
CT' = 2xt and CT == 2y v But these are the intercepts of the tan- 

gent ; therefore, using the formula - -}- ~= 1 , we have — -f- — = 1 

a ° 2x x 2y x 





204 THE HYPERBOLA. 

or xy x -f- yx x = 2xiy v But since 1 > 1 satisfies the equation of the 
curve, x x y x = ?c 2 , and the equation of the tangent may be written 
in the form 

xyx + yx\ = ic 2 . 

This is the equation of a tangent, only when P x is a point of the 
curve, since otherwise the last substitution could not be made. 
The equation of the diameter passing through P x is yx x == xy x , 

of which the direction ratio is — . That of the tangent is — — \ 

Xi ° x x 

Therefore the direction ratios of conjugate diameters as referred to 
the asymptotes are the negatives one of the other. 

280. The conjugate hyperbola has the same asymptotes and the 
same value of c. But one of the co-ordinates of any point of it 
would be negative and the other positive ; hence its equation is 

xy = — \c 2 . 

Therefore the equations of conjugate hyperbolas, thus referred, differ 
in the same way as when they are referred to conjugate diameters — 
namely, in the sign of the absolute term. Moreover, if we put zero 
in place of the absolute term, we shall have the equation of the 
asymptotes. For the equations of the axes are x = and y = 0, 
and their compound equation is xy = 0. 

The shape of each of the hyperbolas xy= ± \c 2 is determined 
solely by the inclination of the co-ordinate axes. They are both rec- 
tangular or equilateral if the axes are rectangular. 

The axes of the curves are the lines x=y and x = — y, which 
are perpendicular, because they bisect the angles between the co- 
ordinate axes, whatever be their inclination. 

281. If the equation of a straight line, y == mx -j- b, be com- 
bined with xy = ic 2 , we shall find for the intersections, 

y = lb ± Wc*m + 6 2 , x = — l-±— y/fm _|_ 52 

From these values may be derived the equation 
y = mx i cy — m, 



HYPERBOLA REFERRED TO ITS ASYMPTOTES. 205 

for the tangent having a given direction, and the equation y = — mx, 
for a diameter bisecting chords parallel to y = mx. 

A line of the form x = a, or one of the form y = b, evidently 
intersects the curve in only one point; therefore, as before shown, 
lines parallel to either asymptote cut the curve in a single point. 
By combining either of these equations with the equations of two 
conjugate hyperbolas, it will be found that a line parallel to an 
asymptote cuts conjugate hyperbolas in points equally distant from 
the other asymptote. 

The ordinate of the middle point of a chord is ib ; that is, half 
the intercept on the axis of Y. Hence this point is equidistant 
from the points in which the secant line cuts the axes, as well as 
equidistant from the points in which it cuts the curve. It follows, 
that the portions of the secant, intercepted between the curve and the 
asymptotes, are equal. 

282. The equations xy = ± |c 2 are central equations. There- 
fore, putting x — x' and y — y in place of x and y, we find 

x y — y' x — x 'y + x V + %° 2 — o> 

for the equation of an hyperbola with centre at P' and asymptotes 
parallel to the co-ordinate axes. Hence, 

Bxy -f Bx -f Ey + F = 

is the general equation of the hyperbola referred to axes parallel 
to its asymptotes. 

Comparing these equations, we find 

E , D , .; DE — FB 

x== , y = — — , ±ic 2 = . 

B B B 2 

By these values we may determine the centre and distance of the 
foci, when the equation is given in its general form. The asymp- 
totes and axes of the curve may now be drawn, and their inclina- 
tions together with the value of c determine the semi-axes. 

If DE — FB is positive, the transverse axis is parallel to the 
straight line x=y, which bisects the angle between the positive 
directions of the co-ordinate axes. If it is negative, the transverse 
axis is parallel to x = — y. The condition that the equation 
shall represent straight lines is 
18 



206 THE HYPEEBOLA. 

DE — FB = 0. 

Examples. — Determine the equation by the conditions that the 
curve shall pass through (1, 2), (2, 9) and (— 1, 0); and give the 
equation of its transverse axis. 

What is represented by 2xy — 4cc -f- 3y — 6 — 0? 

283. The equation of the tangent at P x to the hyperbola 
xy = ± ic 2 , is xy x J r yx 1 = I t k 2 . Hence, for the hyperbola 
whose centre is at F, we have, by substituting x — x' for x, x x — x* 
for x Xl etc., 

ayi + 2/ x i — 3/ O + «i) — d {V + j/i) + 2 (xy q= | c 2 ) = 0. 

Then, substituting for — /, — x f and x'y f q= ic 2 their general values 

D E F 

— , — and — , found by comparing the equations of the last Article, 
B B B 

we have 

£B (xy^yx,) + JD (x + i$ + £E (y + yj + F = 0. 

This is a formula for the tangent to Bxy -f- Dx -J- Ey -|- F = at 
the point P 2 on the curve. If P 2 satisfies this equation, it is con- 
nected with P x by a reciprocal relation, so that P x and P 2 may be 
defined as points reciprocally polar. But it is shown in Art. 277, 
that this reciprocal property, and the fact that the polar of a point 
on the curve is the tangent at that point, furnish geometrical 
constructions of the polar of a point which are independent of the 
position of the co-ordinate axes. Therefore the equation is in 
general that of the polar of P l5 with respect to the hyperbola 
Bxy -f Dx + Ey + F = .0. 



CHAPTER VIII. 

GENERAL EQUATION OF THE SECOND DEGREE. 

284. The most general equation of the second degree, between 
the co-ordinates x and y, is 

Ax 2 -f Bxy -f Cy 2 -f Bx -f Ey + F = 0, 

in which the coefficients and absolute term are constants which may 
have any values. It is the object of this Chapter to investigate, in a 
general manner, the locus or line represented by an equation of 
this form. 

For this purpose we shall use the method of " arbitrary trans- 
formation" referred to in Art. 98 ; that is, we shall use the formulae 
of transformation, regarding the constants introduced by them into 
the equation as arbitrary quantities to be determined in such a 
manner as to simplify the equation as much as possible. The 
transformations used are of two kinds ; the first producing a change 
in the position of the origin, the second a change in the direction 
of the axes. The arbitrary quantities are the co-ordinates of the 
new origin, and the inclinations of the new axes. If they can be 
so determined as to make certain terms disappear from the trans- 
formed equation, it will take forms with which we are already 
familiar. Thus, if the term containing xy can be made to disappear, 
the equation will be known to represent a conic section ; if the 
terms containing x 2 and y 2 can be made to disappear at once, it 
must represent an hyperbola, whose asymptotes are parallel to the 
new axes. 

285. The general equation includes the equations of all conic 
sections ; for, as noticed in Art. 265, the term containing xy may 
be introduced by transformation to axes having new directions. We 
shall therefore, first establish a criterion by which we may dis- 
tinguish between the equations of the ellipse, parabola and hyper- 

207 



208 GENERAL EQUATION OF THE SECOND DEGREE. 

bola, when given in the general form. This criterion will be 
furnished by the equation which gives the intersection of the 
curve by a straight line. For we have seen that there are two 
directions in which if a straight line be drawn it will cut the hyper- 
bola in a single point, while for the parabola these directions coin- 
cide, and for the ellipse they do not exist. As in Art. 237, these 
directions will be found by the condition that the equation by which 
the intersection is found shall reduce to the first degree. 

Now if we substitute y = mx -f~ h in the general equation, we 
shall have a quadratic equation for x, in which the terms of the 
second degree are 

(A -J- Bm -f Cm 2 )f. 

Therefore the condition that a straight line shall cut the curve in 
a single point, is A -f- Bm -j- Cm 2 = 0,* or solving the equation, 

— B ± i/B 2 — 4AC 



m s= 



2C 



If the quantity under the radical sign is positive, there are two 
values of m satisfying the condition, if it is zero there is but one, 
and if it is negative there are none. Hence, 

for the hyperbola, B 2 — 4AC > y 

for the parabola, B 2 — 4AC = ; 

for the ellipse, B 2 — 4AC < 0. 

The equation of the hyperbola, in Art. 282, satisfies the above 
condition, because B 2 is essentially positive, and A and C, in that 
case, are each equal to zero. When B — 0, we find, as in the last 
Chapter, that for the hyperbola A and C must be of opposite signs; 
for the ellipse they must be of the same sign, and for the parabola 
one of them must be zero. 

* When the coefficient of the highest terra in an equation vanishes, so 
that its degree is reduced, in a special case, the root which disappears be- 
comes infinite. Thus, given the equation ax 2 -\- bx -\- c —= ; put x = , and 

we have a -(- bz -f- cz 2 = to determine z. When a vanishes, one value of 
2 is zero, therefore one value of z is infinite. This is, therefore, the general 
method of ascertaining whether a curve has infinite branches. 



CHANGE OF ORIGIN. 209 

286. The equation of a straight line passing through the origin 

is of the form y = mx. When the values of ra, found in the last 

Article, are possible, there are two lines of this form which cut the 

V 
curve in a single point. If we put - for ra in the equation 

A -J- Bra -j- Cra 2 = from which the values of ra were derived, we 
shall have an equation satisfied by any point on either of these 
lines. Making the substitution and clearing of fractions, we find 

Ax 2 + Bxy + Cy 2 = 0, 

which is therefore the compound equation of the two lines. In 
other words, if we place the terms of the second degree equal to 
zero, we shall have the equation of a pair of straight lines, each of 
which cuts the curve in a single point. Their direction ratios are 
the values of ra in the last Article. They will be real straight 
lines for any equation representing an hyperbola, but imaginary 
for one representing an ellipse. 

In an equation representing a parabola the terms of the second 
degree will form a complete square. For if B 2 = 4 AC, we have 

m = — — = , and the expression C (y ■ — mx) 2 when ex- 

20 B 

panded may be reduced to Ax 2 -f- Bxy -f- Cy 2 . In this case 
Ax 2 -f- Bxy -f- Cy 2 === will represent two coincident lines of the 
form y = mx. When B '== and A = 0, this equation becomes 
Cy 2 = or y 2 == 0, representing lines coincident with the axis of 
X ; when B — and C = 0, it becomes Ace 2 = or x 2 = 0, repre- 
senting lines coincident with the axis of Y. 

Examples. — Determine the straight lines represented by each 
of the following equations : x 2 — 3xy -J- 2y 2 = 0, 4x 2 -\- Axy -\-y 2 = 0, 
2x 2 — xy = 0, x 2 — xy -f- y 2 = 0, ra (x 2 — y 2 ) 4- (ra 2 — Y)xy= 0. 

Change of Origin. 

287. We shall now proceed to apply the formulae of transforma- 
tion, and first, those of Art. 85, for passing to a new origin, which are 

x = X + x' and y == Y +f, 

in which x' and i/ are the co-ordinates of the new origin, 
18* 



210 GENERAL EQUATION OF THE SECOND DEGREE. 

Substituting these values in the general equation, and denoting 
the coefficients of corresponding terms by the same letters, we shall 
find that the values of A, B and C are unchanged, and that the 
new values of D, E and F are 

D' == 2Ax' + By + B, E' =2Cy + Bx f + E 

and F == Ax' 2 + Bx'y' + Cy 2 + Bx' + Ey + F. 

Hence, transformation to a new origin does not affect the terms of 
the second degree, but may be used to make two of the other terms 
vanish. It is evident that F' will be zero when the new origin 
satisfies the given equation. But the most symmetrical, as well as 
the simplest result, is obtained by making W = and E' = 0, 
which gives two equations of the first degree, for x' and y , 

2 Ax' + By + D = and 2Cy + Bx' + E = 0. 

Solving these equations we find 

2CD — BE . , 2AE — BD 

and y = 



B 2 — 4AC B 2 — 4 AC 

for the co-ordinates of the point to which the origin must be trans- 
ferred, in order that the terms of the first degree may vanish from 
the equation. 

Examples. — To what origin must we transform 2x 2 -j- 3xy -j- 
y 2 -f- 2x — 5y -f- 6 = 0, in order to make the first powers of x and 
y disappear ? Find the corresponding value of F', and verify by 
actual transformation. 

288. If B 2 — 4AC = 0, these values of x' and y' are infinite ; 
in that case, therefore, it is impossible to reduce the equation to 
the form Ax 2 -f Bxy + Cy 2 + F = 0. If B = 0, the values 
of x' and y' reduce to those of the co-ordinates of the centre, in 
Art. 263, and if both A = and C === 0, they reduce to those of 
Art. 282. 

By means of the values of D', E' and F', we may compute the 
coefficients of the equation for any new 'origin directly by substitu- 
tion of the given values of x' and y f . We may use the calculated 
values of D' and E' to assist in finding that of F ; for DV -f E'?/ = 
2Ax' 2 -f 2Bx'y' -j- 2Cy 2 + Bx' -f Ey, to which it is only neces- 
sary to add Bx' -p Ey -f- 2F to make it the value of 2F'. Hence 



CHANGE OF ORIGIN. 211 

F = i (D' + D) x' + i (E' -f E)/ -f F. 

This expression for F' may be used after we have found the values 
of D' and E'. 

Examples. — What does the equation 3x 2 — 2xy -\-y 2 — 5a: + 
2y — 9 = become, when the origin is transferred to (2, — 1) ? 
to (0, 6) ? to (3, 3) ? 

What when reduced to the form Ax 2 -f Bxy + Cy 2 -f F' = ? 

289. The value of F' in the reduced form 

Ax 2 + B.ry + Cy 2 + F = 

is readily computed by the formula in the last Article, because D' 
and E' are, in this case, each equal zero. To express F' in terms of 
the coefficients, substitute for x' and y' the values found in Art. 287. 
The result is 

}D(2CJ)-:BE) + iE(2AE-BD) 
B 2 — 4AC 

AE 2 + CD 2 -fFB 2 — BDE — 4ACF 
B 2 — 4AC 

When the numerator of this value is zero, the transformed equation 
takes the form Ax 2 -f- ~Bxy -f- Qy 2 = 0, which we found, in Art. 286, 
to represent two real or imaginary straight lines. Therefore 

AE 2 + CD 2 -f FB 2 — BDE — 4 ACF = 

is the condition which must be fulfilled by the coefficients when the 
equation represents straight lines. If B = 0, this condition re- 
duces to that of Art. 261 ; if A = and C = 0, it reduces to that 
of Art. 282. 

Examples. — What value must be given to A, in order to make 
Ax 2 -f- xy -f- 2y 2 -\- x — 5y -j- 2 = represent a pair of straight 
lines? (Substitute the values of the given coefficients in the con- 
dition and determine that of A.) 

Determine the values of B. for which x 2 -\- 3xy -j- 2y 2 -\- y — 1 = 
represents straight lines ; and reduce each of the resulting equations 
to the form Ax 2 -f- Bxy -\- Cy 2 = by transformation. 

Form the compound equation of the straight lines 2x = 3y — 1 
and x -f- y = 2 (see Art. 81), and show that it satisfies the above 



212 GENERAL EQUATION OF THE SECOND DEGREE. 

condition; also that the point (x f , y f ) determined by the formulae 
of Art. 287, is the intersection of the given lines. 

Change in Direction of Axes. 

290. We shall apply the formulae for a change in the direction 
of the axes to the terms of the second degree only, in the general 
equation, for we have seen that these terms are not affected by a 
change in the position of the origin. The formulae (Art. 90) are 

X sin (at — a) -j- Y sin (at — 0) 

cc ^^ ■ 

sin at 

X sin a -f Y sin 

y— = > 

sin at 

in which to denotes the angle between the old axes, and is therefore 
fixed, while a and denote the inclinations of the new axes to the 
old axis of X, and are therefore arbitrary. 

Before making the general application of these formulae, we shall 
consider the case in which the axis of Y only is changed. The 
value of a will then4)e zero, and the formulae reduce to 

sin at sin at 

Substituting these values for x and y, in Ax 2 -j- Basy -j- Qy 2 , and 
denoting the new coefficients by A', B' and C, we find 

A' = A E , ^_ 2A sin (at — g) -f B sin g 



A sin 2 (at — 0) + B sin (at — g) sin /5 + C sin 2 



The coefficient of x 2 is therefore unchanged by this transformation. 

291. To make B' == 0, we must give to /? such a value that 

sin 3 2A 

2A sin (at — fi) 4- B sin = 0, or ( = . Now 

sin (at — /3) B 

— expresses the direction ratio of the new axis of Y. For 

sin (at — 0) 

let y =mx -\- b be the equation, as referred to the old axes, of a 



CHANGE IN DIRECTION OF AXES. 213 

line parallel to the new axis of Y ) then we know that the trans- 
formed equation of this line will be of the form X = a. Making 
the transformation, we have 

*>/»-»>»" ("-Dy-wX -6 = 0, 
sin to 

in which, therefore, the coefficient of Y must equal zero. Hence 

= m, the old direction ratio of the line, or of the new 



sin (to — /9) 

axis of Y, which is parallel to it. Therefore we shall have B' = 0, 
or the term containing xy will vanish from the equation of the 
second degree, if the axis of Y be made parallel to a line whose 

2A 

direction ratio is , the axis of X being unchanged. 

The above expression for the direction ratio of a line, in terms 
of its inclination to the axis of X, will be frequently used in the 
following Articles. The direction ratio of the new axis of X, 

whose inclination is a in the more general formulae, is— ; — . 

sin (tt> — a) 

292. Since the term containing xy may thus always be made to 
vanish by transformation, it appears that every equation of the second 
degree must represent a conic section. The value of the quantity 
B 2 — 4AC determines to which of the three classes of conies the 
locus of a given equation belongs ; since, by Art. 285, it is positive 
for an hyperbola, zero for a parabola and negative for an ellipse. 
To show that the sign of this quantity is unchanged by transforma- 
tion, we find the value of B' 2 — 4 A'C\ Substituting the values of 

A', B' and 0', and reducing, we find : — ■ — for the 

sin 2 (o 
value of this expression. Therefore 

B' 2 — 4A'C B 2 — 4AC 

sin 2 /3 sin 2 u) 

Since the direction of the axis of X has not been changed by 
this transformation, /?, which denotes the inclination of the new 
axis of Y, is the angle between the new axes, while io is the angle 

j$2 4AC 

between the old. Therefore the value of the quantity 

sin 2 m 



214 GENERAL EQUATION OF THE SECOND DEGREE. 

is not affected by a change in the direction of the axis of Y. In 
the same manner, it may be shown to remain unaltered when the 
axis of X is changed. Now we may transform an equation to new 
axes passing through the same origin, by making the required 
change, first in the axis of Y, and then in that of X. The result 
will be the same as if we had made the transformation at once by 
the general formulae ; for the equations found by the two methods 
will represent the same line, and will contain the same absolute 
term, and therefore must be identical. Therefore in general, the 

value of the quantity ; is unchanged by transformation of 

sin 2 at 

co-ordinates. 

Since the denominator of this quantity is essentially positive, the 
numerator can never change sign. 

Examples. — Determine the class to which each of the following 
conies belongs : 2x 2 — xy — y 2 -\- §x -\- y — 5 = 0, Zxy — y 2 -{- 
7x = 0, 2x 2 — 4:xy + 2y 2 — x = 0. 

Transform the first, from axes making an angle of 60°, to axes 

j>2 4AC 

bisecting their angles, and find the value of — ; both before 

sin 2 at 

and after the transformation. (The formulae for this transforma- 
tion are given at the end of Art. 90.) 

293. We now substitute the general values of x and y in the 
expression Ax 2 -j- ~Bxy -j- Cy 2 , and, denoting the new co-enicients 
by A', B' and C, we find 

f A sin 2 (a* — a) -f- B sin (at — a) sin a -f- C sin 2 a 



_ 2A sin (to — a) sin (to — /3) + B sin (01 — aS pin /3 + R sin Ca> — 0) sin a -f 2C sin a sin 

sin- «> 

_ A sin 2 (w — g) -f B sin (at — ft) sin j3 -f- C sin 2 ft 
sin 2 at 

The value of A' involves the angle a only, and that of C involves 
the angle ft only. Therefore the coefficient of x 2 , for a given conic, 
is independent of the direction of the axis of Y, and that of y 2 is 
independent of the direction of the axis of X. 

294. To make A' = 0, we must have A sin 2 (at — a) -j- 
B sin (at — a) sin a -j- C sin 2 a = 0, or, dividing by sin 2 (at — a) 



CHANGE IN DIRECTION OF AXES. 215 

and denoting , which is the direction ratio of the new 

sin (to — a) 

axis of X, by m, 

A -f Bm 4- Cm 2 == 0. 

This is the same quadratic for m which we found, in Art. 285, for 
the direction ratios of straight lines which cut the curve in a single 
point. Since the resulting values of m are imaginary when 
B 2 — 4AC is negative, it is impossible to make the term contain- 
ing x 2 vanish from the equation of an ellipse. For the hyperbola, 
this term will vanish when the axis of X is parallel to either asymp- 
tote, and for the parabola, it will vanish when this axis is parallel 
to the axis of the curve. 

The condition C = gives the same equation to determine the 
direction ratio of the axis of Y. Since there are two values of m 
which satisfy the equation when B 2 — 4AC is positive, both x 2 and y 2 
can be made to disappear from the equation of an hyperbola. In 
the case of the parabola, either one of these terms can be made to 
vanish ; but only one, because the condition is satisfied by but one 
value of m, and the co-ordinate axes cannot have the same direction. 

Examples. — Determine, for each of the conies in the examples 
under Art. 292, the direction ratios of the new axes, for which 
x 2 and y 2 will disappear from the equations. 

295. Finally, to make the term containing xy disappear (and 
thus reduce the equation to the form discussed in the last Chapter), 
we must have B' = 0, or 

2A sin (w — a) sin (a> — 0) + B sin (w — a) sin + B sin (to — 0) sin a + 2C sin a sin =0. 

This equation, therefore, expresses a condition which is fulfilled by 

a and /?, when the new axes are parallel to a pair of conjugate 

diameters. It is, then, the general relation between the inclinations 

of conjugate diameters. To express it in the form of a relation 

between direction ratios, divide by sin (u> — a) sin (w — /5), and 

j sin a _ sin B '"' m 

denote and by m and m . The result is 

sin (w — a) sin (cu — /S) 

2A + B (m -f m') 4- 2Cmm f == 0. 
This is the general relation between the direction ratios of conju- 



216 GENERAL EQUATION OF THE SECOND DEGREE. 

gate diameters as referred to any co-ordinate axes. Making B = 0, 
this equation expresses that the product of these direction ratios is 
constant, when the co-ordinate axes are parallel toa pair of conjugate 
diameters. Making A = and C = 0, it expresses that the direc- 
tion ratios are the negatives one of the other, when the axes are 
parallel to the asymptotes. Compare Arts. 205 and 279. 

The equation also shows that each asymptote is conjugate to itself; 
for, making m! = m, we have the equation A -}- Bm -j- Cm 2 = 
to determine the direction ratio of a line conjugate to itself. But 
this is the equation whose roots are the direction ratios of the 
asymptotes. 

Examples. : — Find the relation between the direction ratios of 
the conjugate diameters of 2x 2 — Sxy -j- 5y 2 -j- 2x — y -j- 4 = 0, 
of x 2 — xy -j- y 2 — x -f- y = 0. 

The Central Equation. 

296. We are now prepared to examine the forms to which it is 
possible to reduce the equation of any conic section by transforma- 
tion of co-ordinates, and the relations between the curve and the 
co-ordinate axes, which are impHed by the form of the equation. 

In Art. 287, we found that by change of origin it is generally 
possible to reduce a given equation to the form 

Ax 2 + Bxy -f Cy 2 -fF = 0. 

The new origin is the centre of the curve ; for the term Bxy can 
now be made to vanish by a change in the direction of the axes, 
and then the equation will take the form of the central equation 
either of the ellipse or of the hyperbola. The general co-ordinates 
of the centre were found, in Art. 287, by means of two equations, 
which express that it is on each of the lines 

2Ax -f By + D = and 2Cy + Bx -f E = 0. 

Therefore these lines are generally diameters, whose intersection is 
the centre. 

297. In the case of the parabola these lines are parallel, but they 

.„ ,. o , . ,. . . 2A j B 

are still diameters; for their direction ratios are — — and — — ; 

B 2C 



THE CENTRAL EQUATION. 217 

but when B 2 — 4AC = 0, these quantities are equal, and by Art. 286, 
each of them expresses the direction ratio of the lines which cut the 
parabola in a single point. By the principle of combined equations, 

2Ax -fB^ + D + i (2Cy -f Bx + E)-= 

will represent a series of lines passing through the intersection of 
these diameters in the cases of the ellipse and hyperbola, or parallel 
to them in that of the parabola. Hence it is the general equation 
of a diameter to the given conic. 

The ellipse and the hyperbola, for which the diameters intersect, 
are sometimes called central curves, in distinction from the parabola, 
for which the diameters are parallel. Since the co-ordinates of the 
centre become infinite for the parabola, it is generally impossible to 
reduce a given equation to the central form, if B 2 — 4 AC = 0. 

293. The reduction, however, may be made, if the coefficients of 
the given equation fulfil the condition 2CD — BE = 0, as well 
as B 2 — 4 AC = 0. For these two conditions will make the equa- 
tions 2Ax -f By -f D = and 2Cy -f Bx -j- E = represent the 
same line. In this case all the diameters will be coincident, and the 
general co-ordinatey of the centre take the indeterminate form. The 
centre may therefore be taken anywhere upon the single diameter. 
The numerator of the expression for F', in Art. 289, in which 
general co-ordinates of the centre were used, now becomes zero. 
Therefore the condition which generally makes F' = 0, and shows 
that the equation represents straight lines, is fulfilled in this case, 
but the expression for F' is indeterminate. Its value may, however, 
be found by substituting the co-ordinates of the assumed centre in 
the expression in Art. 287, and thus the equation will be reduced 
to the central form, Ax 2 + Bxy + Cf + F' = 0. 

299. When the central equation fulfils the condition of the 
parabola, B 2 — 4 AC = 0, it may be written in the form 



co — mx) 2 + r=0, 
2C~ B 



B 2A 

in which m = = — — . Compare Art. 286. This is 

2C B F 

/ w 

equivalent to y = ma;±:-v/ , and therefore represents two 

19 K 



218 GENERAL EQUATION OF THE SECOND DEGREE. 

parallel lines equally distant from y = mx. Therefore the equa- 
tion of a true parabola cannot be expressed in the central form. 
But the equation of a pair of parallel lines fulfils the general con- 
dition of the parabola; and such lines constitute a conic, all of whose 
diameters coincide with the line midway between the parallels, and 
whose centre is any point of that line. 

If F' and C are of the same sign, the parallel lines are imaginary, 
because the radical in the value of y is then imaginary. 

300. The intercepts of Ax 2 -f- Bxy -f- Cy 2 -j- F = are 

I F / F 

x == ±: vj , y = zhV— — . If F' = 0, the intercepts are 

all zero, and the equation represents two real coincident or imagin- 
ary lines, as shown in Art. 286. Two intersecting straight lines, 
therefore, constitute a conic of which the point of intersection is 
the centre. The imaginary straight lines may be regarded as 
having a real point of intersection at the centre, which is in fact 
the only real point which satisfies the equation. Their separate 
equations are of the form y = mx, having imaginary values of m ; 
but the equations of the imaginary parallel lines referred to in the 
last Article are of the form y = mx -j- b, having real and equal 
values of m, but imaginary values of b. 

Since for the ellipse B 2 — 4AC is negative, while B 2 is essentially 
positive, A and C must have the same sign. If F' has the opposite 
sign, the intercepts are both real and the ellipse is real; but if F' 
has the same sign, the intercepts are imaginary, and the equation 
has no locus ; that is, it is satisfied by no real points. 

If B 2 — 4AC ]> 0, A and C may have opposite signs; therefore, 
for the hyperbola, either or both pairs of intercepts may be imaginary; 
and since A or C may be zero, either or both may become infinite. 

Examples. — Reduce to the central form and discuss the equa- 
tions, 2x 2 — Sxy -f- By 2 -f 3.x — 6> = 0, 2x 2 — xy + y 2 -f 3x — 
hy -f 4 = 0, 9x 2 + 6xy + y 2 -f 12.x -f 4y -f- 5 = 0, x 2 -f xy — 
if -f 3^ = 0. 

301. The central equation may be further simplified by thb 
proper changes in the direction of the axes. Thus, in Art. 291, 
we found that, by making the axis of Y parallel to a line whose 

" '"''■ : ~ 2A 

direction ratio is , the term containing xy will be made L o 

B 






THE CENTRAL EQUATION. 219 

vanish. The diameter 2Ax + By -f- D = (see Art. 296) has 
this direction ratio. Therefore if the equation of a conic be re- 
ferred to this diameter as axis of Y, and a diameter parallel to the 
axis of X, it will take the form 

av + cy + F == 0. 

Hence 2Ar» -j- By -J- D = is the diameter conjugate to that 
parallel to the axis of X. In like manner, it might be shown that 
2Cy -f- Bx -J- E = is the diameter conjugate to that parallel to 
the axis of Y. But, by Art. 295, B' = whenever the axes are 
made to coincide with lines whose direction ratios are connected by 
the relation 2 A -j- B (m -f- m') -j- 2Cmm f = 0. Therefore in gene- 
ral, the equation of a conic will take the above form, when referred 
to the lines 

y — y = m (x — a/) and y — y' = m f (x — a/), 

in which x' and y' are the co-ordinates of the centre, if m and mf 
satisfy this relation ; that is, if 

2A 4- Bm 

m = — ; 

B+ 2Cm 

hence, when this condition is fulfilled, the lines are conjugate diame- 
ters. Since the value of m may be assumed at pleasure, only 
three conditions are thus imposed on the new axes, and only three 
of the terms of the general equation have been made to disappear. 

2A B 

If m = — — , we find m' = 0, and if m = — — we find 
B 2C 

m' = oo. These results agree with what is shown above respecting 
the diameters of Art. 296. 

302. In Art. 294, it was shown that the term containing x 2 will 
disappear when the axis of X is parallel to an asymptote. In a 
similar manner the term containing y 2 can be made to disappear. 
Therefore if a conic be referred to its centre and an asymptote, its 
equation will take the form 

B'xy + Cy -f F' = 0, or AV -f- B'xy -f F = 0, 

according as the asymptote is made the axis of X or that of Y. 



220 GENERAL EQUATION OF THE SECOND DEGREE. 

The equation y — y' = m (x — x'~) represents an asymptote when 
m is one of the roots of A -J- 3m -\- Cm 2 = 0. 

The term B'xy cannot be made to disappear when one axis is an 
asymptote ) because, as shown in Art 295, if m has one of these 
values, we shall find m' == m ; but the co-ordinate axes must inter- 
sect, and therefore must have different direction ratios. 

If the two asymptotes are taken as axes, the equation will take 
the form 

B'^ + F' = 0, 

and this is the only way in which four terms of the general equa- 
tion can be made to disappear. 

The equations of hyperbolas only can be expressed in the forms 
of this Article. If F' === 0, the first equation may be written in 
the form (B-ap -f- Q'y^)y = 0, which represents the two straight lines 
B'x -J- C'y === and y = 0, the latter being the axis of X. Under 
the same supposition, the second equation becomes x (A'sc-j-B'y)— 0, 
which represents two straight lines, of which one is x = 0, the 
axis of Y ; and the third equation becomes xy = 0, which repre- 
sents both axes. 

The Conic Referred to a Tangent. 

303. Another method of simplifying the equation of a given 
conic, by transformation, is to make the absolute term and one of the 
terms of the first degree disappear. It is not always possible to do 
this by a change of origin only ; for, if we put the values of D' and F' 
(Art. 287), or those of E' and F', equal to zero, we shall have two 
equations which, since one of them is of the second degree, may 
give imaginary values of x' and y' . However, the absolute term 
will disappear, if the new origin be taken at any point of the curve; 
and we shall find that one of the terms of the first degree may be 
made to vanish by giving a proper direction to one of the axes. 

Let P x denote a point on the given conic; then, denoting the new 
values of D and B by J) 1 and E 1} the equation referred to P x as 
origin is 

Ax 2 -f Bxy -f Cy 2 + D,x +% = 0; 

for, referring to Art. 287, the new value of F is zero, because P x 
satisfies the given equation. By the same Article we have 



THE CONIC REFERRED TO A TANGENT. 221 

D x = %hx t + B 9i + D and E x = 2C^ -f Bx x -j- E. 

It is evident that the equation of an imaginary ellipse, which is 
satisfied by no real points, cannot be put in the above form, which 
represents only real loci passing through the origin. 

If J) l = 0, the new origin is on the diameter 2 Ax -f By -j- D = 0, 
because its co-ordinates satisfy this equation. In like manner, if 
E x = 0, it is on the diameter 2Cy -j- Bx -f E = 0. 

304. "We may now make the term containing y disappear, by 
changing the direction of the axis of Y. Using the simpler formulae 
of Art. 290, the terms of the second degree receive the new coeffi- 
cients of that Article, but do not give rise to new terms of the first 
degree. We have now to examine the effect of this transformation 
on the terms of the first degree. 

In Art. 291, we saw that the coefficient of y will vanish from the 
equation of a straight line, y = mx -\- b. if the axis of Y be made 
parallel to it ; that is, coincident with y = mx. Now D^ -j- Ej y = 
is the equation of a straight line passing through the origin. There- 
fore, if the axis of Y be turned until it coincide with this line, the 
terms of the first degree, D x x -(- E lt y, will reduce to Djrr, and the 
equation will take the form 

Ax 2 + B'xy -f cy + IV = 0. 

The coefficient of x is unchanged by this transformation, as it is in 
the example of Art. 291. 

In an equation of this form, the axis of Y is a tangent to the 
curve at the origin. For, making x = 0, we find y 2 = 0, or both 
values of the intercept on the axis of Y become zero, therefore this 
axis meets the curve at the origin, in Hco coincident points. 

305. We see then that the straight line 

J) x x + E^ = 

is tangent to Ax 2 -j- Bxy -\- Cy 2 + J) x x -j- E^ = 0, at the origin. 
That is, if, in a given equation containing no absolute term, ive put 
the terms of the first degree equal to zero, the result will be the equa- 
tion of a tangent to the curve at the origin. Accordingly, for an 
equation of the form given in the last Article, the tangent is the 
line D x ^ = or x — 0, the axis of Y. If the axis of X had been 

19* 



222 GENERAL EQUATION OF THE SECOND DEGREE. 

made to coincide with the tangent, the equation would have taken 
the form A'x 2 -j- B'xy -f- Cy 2 -j- Yj x y = 0, and y = would have 
been the new equation of the tangent. 

It is impossible to make both of the terms of the first degree dis- 
appear in this manner, for that would require the two axes to coin- 
cide. But in an equation of the form Ax- 2 -j- Bxy -|- Cy 2 ~t 0, 
which we have seen represents two real coincident or imaginary 
straight lines passing through the origin, both axes may be con- 
sidered as tangents. In fact, in this case, every line passing 
through the centre or intersection of the straight lines fulfils the 
algebraic condition of tangency, namely, that of meeting the conic 
in two coincident points. 

In the particular case when the conic becomes two coincident 
lines, every straight line which meets it is a tangent. 

Examples. — Give the equations of tangents to f 6x 2 — xy*-\- 
y 2 +$x— y=0,2x 2 — 4ry-f-2?/ 2 — 3x = 0, x 2 — ±xy + 4y 2 = 0, 
x 2 -f xy = x, 2y 2 = 0, (y — bx -f- 2) 2 = 0. 

306. If we now change the direction of the axis of X in 
Ax 2 -j- B'xy -f- Cy 2 -j- D x x =& 0, we may write for the new equa- 
tion 

AV -f B'xy + Cy + D'x = 0. 

The equation remains of the same form, because the axis of T is still 
a tangent at the origin. The coefficient of x takes a new value, 
which we here denote by D\ The values of A', B' and C are 
those of Art. 293, for we proved in Art. 292 that the result of 
changing the direction first of one axis and then of the other is the 
same as that of using the general formulae of transformation. Since 
the axis of Y was made to coincide with D x x -j- E x y — 0, the value 
of the angle /? is fixed, and the direction ratio of the new axis 
ofYis 

& 

m = . & 

Pa 

The value of the angle a, and consequently the direction ratio of 
the new axis of X, is arbitrary. We may therefore generally give 
it a value, m', determined by the relation between m and m', in Art. 
295. The term containing xy can therefore be made to disappear, 
and the equation will then take the form 



TANGENTS AND DIAMETERS. 223 

AV + Cy + JYx = 0. 

When the equation is in this form, the axis of Y is a tangent, 
and the axis of X is a diameter; because every value of x now 
gives two values of y numerically equal and with opposite signs. 

307. The only case in which Ax 2 + Bxy -f Cy 2 + ~D x x -f E,y = 
cannot be reduced to this last form is when m is one of the roots 
of A -f- Bm -f- Cm 2 == ; for, as mentioned in Art. 295, we should 
then find mf = m. In this case, when we turn the axis of Y into 
coincidence with the tangent Djpc -(- E x y = 0, the term containing 
y 2 will disappear (see Art. 294). Therefore the equation will take 
the form 

Ax 2 + B'xy + D x x = 0, . 

no change being made in the axis of X. This may be written in 
the form x (Ax -\- B'y -|~ ^i) = ^> showing that it represents two 
straight lines,* one of which is x = 0, the axis of Y. When the 
axis of X is made tangent, the similar form B'xy -f- Cy 2 -f- E lt y = 
may occur, showing that the given equation represents two straight 
lines, with one of which the axis of X has been made to coincide. 

Tangents and Diameters. 

308. In reducing a given equation to the form Ax 2 -j- Bay -f- 
Oy 2 -f- D x a5 -f- E lt y = 0, only one condition was imposed upon the 
new origin, namely, that it shall satisfy the given equation. Con- 
sequently we have only been able to make three of the terms of the 
general equation vanish by this method. But, since Pj is any 
point of the curve, we may now express the general equation of the 
tangent at any point. After the transformation of Art. 303 we 
found the equation of the tangent at P x to be J) x x -j- E^ == 0. 
Transforming back to the old axes, by substituting x — x x for x 
and y — y x for y, we have for the equation of a tangent (supposing 
P x to be a point of the curve) 

* The condition that m shall satisfy the equation A -j- Bm -j- Cm 2 = is 
Tm nn 2 

A — ^^ + -^7- = 0, or AEi 2 — BDiEi + CD^ =0. Referring to 

Art. 289, it will be seen that, since F = 0, this is the condition for which 
the conic becomes a pair of straight lines. 



224 GENERAL EQUATION OF THE SECOND DEGREE. 

D 1 {x-x 1 )-\-E 1 (y-y l ) = 0. 

Substituting the values of D x and E 1; Art. 303, we find for the 
general equation of the tangent at a given point, P 1? 

(2A*, + B^ + D) (x - x,) + (2C yi + P% + E) (y -^ - 0. 

309. From the value of m, the direction ratio of this line, which 

is m == — -4; we derive D x -j- wE x = 0,or 
Ei 

2A^ -f B^ + D -f m (2Cy x + B^ -f E) = 0. 

This equation expresses that P x is on the straight line 

2Ax + By + D 4- m (2Cy + Bx + E) = 0, 

which, by Art. 297, is a diameter of the curve, m taking the place 
of the arbitrary constant k. We have therefore found the equa- 
tion of the diameter whose vertex is P x , in terms of the direction 
ratio of the tangent at P x . This is the diameter with which the 
axis of X coincides when the equation is reduced to the form 
A'x 2 -f- B'y 2 -)- D'x — 0. Denoting its direction ratio by m', we 
find, by reducing it to the form y == m'x -\- b, 

2A 4- Bm 

iffl' • ! 

B 4- 2Cm 

the same expression which we found in Art. 301 for the direction 
ratio of the conjugate diameter. Therefore the diameter passing 
through Pj is conjugate to that parallel to the tangent at P x . 

In applying these equations to find the tangent and the diameter 
passing through a given point on the curve, the values of J) 1 and 
E x must first be computed and substituted in the equation of the 
tangent ; then the value of the direction ratio of the tangent is to 
be given to k in the general equation of a diameter. Thus, given 
the conic x 2 -\- 2xy — y 2 -f- 3x — 2y -\- 5 == 0, to find the equa- 
tions of the tangent and diameter at the point (1, 3). (which will be 
found to be the point of the curve). We find J) 1 = 11, E 2 = — 6; 
hence the tangent is 11 (x — 1) — 6 (y — 3) = 0, or lice — 6y -{- 



TANGENTS AND DIAMETERS. 225 

7 = 0. Then, since m = y , the diameter is 2x -j- 2y -f- 3 -f- 
U (_ 2y -|- 2x — 2) = 0, or 17 js — by — 2 = 0. These results 
may be verified by showing that the given point satisfies each of 
the equations, and that the values of m and m' satisfy the relation 
which should exist between them. 

Examples. — Find a tangent and a diameter to 2x 2 — xy — 
3y 2 — 2x -f- y — 4 = at the point (3 — 2) j at each of the points 
where the curve cuts the axis of X. 

Find a tangent and diameter to 2x 2 -\- xy — 3y 2 -f- 3x -J- 7y — 
2 = at (— 3, — 1), and at (— 1, 1) ; to x 2 — 2xy -\-y 2 — 2x + 
2y — 3 = at each of the points whose abscissa is 1. 

310. A similar method may be used in finding conjugate diame- 
ters, when the value of m is given. For, putting the value of m in 
place of k in the general equation of a diameter, we can find the 
equation of the conjugate diameter; then, using the value of m! 
determined by the equation thus found, we can find, in a similar man- 
ner, the equation of the diameter whose direction ratio is m. Thus, 
for the conic 2x 2 — xy — Sy 2 — 2x -f- y — 4 = 0, the general 
equation of a diameter, by the formula of Art. 297, is 

4:x —y — 2 -f- k (— 6y — x -f 1) = 0. 

Let it be required to find conjugate diameters, for one of which 
m = 2. First making k= 2, we find for the conjugate diameter 
2x — 13y = 0, from which m! = ^_; then making k = -j^-, in the 
same formula, we find on reducing 50^ — 25y — 24 = 0, in which 
m = 2, the given direction ratio. 

Since the asymptotes are diameters each of which is conjugate to 
itself, the same method enables us to find the equation of an asymp- 
tote, after the proper value of m is found. Thus, given the same 
conic to find the asymptotes. Substituting the given coefficients in 
the equation A -\- Bm -j- Cm 2 = 0, whose roots are the direction 
ratios of the asymptotes, we have 2 — m — 3m 2 = 0, solving 
which we find m = -| or — 1. Putting the first value of m in 
place of k, and reducing, we have 10^ — 15y — 4 = 0, in which 
m = |^, therefore this is the equation of an asymptote. Putting 
k = — 1, we find for the other asymptote, hx -(- 5y — 3 = 0. 

If the given equation represents two intersecting straight lines, 
the equations thus found will be the separate equations of the lines. 



226 GENERAL EQUATION OF THE SECOND DEGREE. 

Examples. — Find the conjugate diameters of 5x 2 — 2xy -f- 
% 2 + 6x — y -j- 12 = 0, of which one is parallel to y = 2x -j- 1 ; 
find the diameters for which m = 1, m = — 1, ra = 0, etc. 

Find the straight lines represented by 2x 2 -\- xy — 3y 2 -J- 3x -j- 
7y — 2 = 0, and verify by forming the compound equation. 

311. The method explained in the last Article may also be used 
when we require the diameter which bisects chords parallel to a given 
line, or which passes through the point of contact of a tangent 
having a given direction ratio, m. If the curve is a parabola the 
value found for m' will be always the same. For, in that case 

B 2A 

B 2 == 4AC, therefore — = — . From this it is evident that what- 
2C B 

ever be the value of m, the value of rhf, Art. 309, is . This 

2C 

is the common direction ratio of all the diameters. If it be sub- 
stituted for m, the value of m! becomes indeterminate. 

To find the equation of a tangent having a given direction, we 
must substitute the value of m for k, in the formula for a diameter. 
This will give the diameter passing through the point of tangency. 
The co-ordinates of its vertex, or intersection with the curve, must 
then be found and substituted in y — y x = m(x — x^). There 
will of course be but one intersection, and but one tangent, when 
the given curve is a parabola. If it is a real ellipse, there will 
always be two ; and if it is an hyperbola, the intersections may be 
imaginary. 

Examples. — Find the diameters of 4x 2 — 4xy -\-y 2 — 2x = 
which bisect chords parallel to y = x, y = — x, y = 2x, etc. 

Find a tangent to this curve parallel to y == 3x. 

Find tangents to x % -j- xy -\- y 1 -j- 2x — 2y — 8 = parallel to 
each of the co-ordinate axes. 



Rectangular Equations. 

312. The shape of a conic and the directions of its axes are 
most easily investigated when the co-ordinate axes are rectangular. 
We shall, therefore, first show how to transform an equation from 
oblique to rectangular co-ordinates, and then apply the formulae for 
change in direction of rectangular axes. 



RECTANGULAR EQUATIONS. 227 

To make the axes rectangular, let /? = 90° in the transformation 
of Art. 290, in which the direction of the axis of X is un- 
changed. Giving this value to /5, the new coefficients become (since 
sin (to — 90°) = — cos w) 

A'=A, B , = -2Acos m + B ) 

sin to 

A cos 2 to — B cos to -)- C 



In these expressions, to is the angle between the old axes, and there 
is no arbitrary constant of transformation ; therefore any condition 
imposed upon the new or rectangular coefficients will give an 
equivalent condition in terms of the general coefficients. 

313. The condition that the general equation shall represent a 
circle may thus be found. For, in order that the rectangular equa- 
tion shall represent a circle, we must have B' = and C = A'. 
(See Art, 107.) The condition B' == gives B = 2A cos to. The 
condition C = A' gives A sin 2 to = A cos 2 to — B cos to -j- C. 
Substituting for B its value derived from the first condition, this 
reduces to A = C ; therefore 

B = 2A cos to and A = C 

are the two conditions which must be fulfilled by every equation 
representing a circle. 

When the equation is reduced to the central form, it is evident 
that the second condition expresses simply that the intercepts on 
the axes are equal. The first condition, taken by itself, expresses 
that the axis of X, which was unchanged by transformation, coin- 
cides with one of the axes of the conic. It is evident that these 
two conditions can be fulfilled at once only by the circle. 

Examples. — When the inclination of the axes is 60° (whose 
cosine = |), what are the conditions for a circle ? 

Give the central equation of the circle when to = 60°. 

Show by the above conditions that for a circle B 2 — 4 AC < 0. 

What is the inclination of the axes when x l -}- -y/Zxy -j- y 2 = B 2 
represents a circle ? 

314. To pass from one system of rectangular axes to another 
having the same origin, we use the formulae of Art. 88, 



223 GENERAL EQUATION OF THE SECOND DEGREE. 

x = X cos a — Y sin a y == Y cos a -(- X sin a, 

in which there is but one arbitrary quantity, a, the inclination of 
the new axis of X to the old. 

Substituting these values, we find, for the new coefficients of the 
terms of the second degree, 

A' = A cos 2 a -j- B sin a cos a -f- C sin 2 a, 

B' == 2 (C — A) sin a cos a -f- B (cos 2 a — sin 2 a), 

C = A sin 2 a — B sin a cos a -f- C cos 2 a. 

By adding the values of A' and C, we obtain A' -j- C = A -f- ; 
that is, the sum of the coefficients A and C is unchanged hy trans- 
formation, when the axes remain rectangular. In Art. 292 it was 

B 2 — 4AC . 

shown that the quantity — — is unchanged by transforma- 

sin 2 io 

tion. When the axes are rectangular, this expression becomes 

simply B 2 — 4AC, since sin 90° = 1. Therefore A -f- C and 

B 2 — 4AC are two functions of the coefficients, whose values are 

unaffected by this transformation. 

315. In order to find the direction of an axis, of the curve, we 

have to find that value of a which makes B' = 0. By the formulae 

for the sine and cosine of the double angle, sin 2a = 2 sin a cos a 

and cos 2a = cos 2 a — sin 2 a. Therefore the condition B' = gives 

(A — C) sin 2a = B cos 2a. 

This result may also be found without the aid of trigonometrical 
analysis, as follows : Since the semi-diameters inclined at equal 
angles on each side of the major or transverse axis are equal, the 
lines bisecting the angles between equal diameters are the axes of 
the curve. Suppose the equation to be central, the intercept on 

the axis of X is -v/ . Hence if A' = A, the intercept on the 

new axis of X will be equal to that on the old. Putting A = A' 
we derive 

(A — C) sin 2 a = B sin a cos a, 

in which a denotes the inclination of a diameter equal to that 



RECTANGULAR EQUATIONS. 229 

measured on the axis of X. This equation is satisfied by making 
sin a = or a — 0°, which makes the diameter coincident with 
the axis of X, and also by making (A — C) sin a = B cos a, in 
which, therefore, a is twice the inclination of an axis of the curve. 
This is the same equation for a as that given above for 2a, in which 
a is the inclination of the axis. Hence when a denotes the inclina- 
tion of an axis, we have 

tan 2a 



A — C 



Since the angles 2a and 180° -f- 2a have the same tangent, the 
two angles a and 90° -\- a (of which we may suppose a to be in 
the first quadrant) satisfy this equation. The inclinations of both 
axes are thus given by the same equation. If B == 0, the inclina- 
tions are 0° and 90°, because the axis of X already coincides with 
one axis. If A = C, tan 2a becomes infinite, 2a = 90° and 
a = 45° ; unless at the same time B = 0, which makes the curve a 
circle and the direction of the axes indeterminate. 

Examples. — The co-ordinate axes being rectangular, give the 
inclinations of the axes of Sx 2 -J- 2xy -{- y 2 -f- 2x — y -f- 3 = ; 
of x 2 -j- xy -f- y 2 = 3 — x ) of ^ — 2xy -|- y 2 = 8 \ of xy -\- 
y 2 = Q. 

316. When the co-ordinate axes are oblique, the value of tan 2a 
may be found by substituting in the above expression the values 
of A', B' and C in Art. 312 ; for these are the coefficients in the 
rectangular equation, when referred to the same axis of X. 
Hence 

B' (B — 2 A cos w) sin w 



tan 2a == 



C A sin 2 to — A cos 2 a> -f- B cos w — C 



This is the general formula for the inclination of the axes of the 
curve when w, the inclination of the co-ordinate axes, is known. Thus 
if w = 60°, so that cos a) = J, and sin w = J-j/3, we have in gene- 
ral tan 2a = ^ ' * ■ . Therefore in the given equation 

A — {— x> — LKj 

3x 2 -f 2xy + 2# 2 = 36, tan 2a = — j/3, 2a == 120° and a = 60° ; 

that is, one axis of the curve coincides with the axis of Y. 

By making use of the formulae for the double anr^le, referred to 
20 



230 GENERAL EQUATION OF THE SECOND DEGREE. 

in the last Article, the expression for tan 2a may be put in the 
following form : 

A sin 2a> — B sin to 

tan la = -. 

A cos 2w — B cos id -j-C 

Examples. — If u> = 45° , what are the inclinations of the axes 
of the conic x 2 — xy -j- y 2 -J- 2x — 2y -}- 1 == ? 

Show generally, that if A = C, the axes of the curve bisect the 
angles between the co-ordinate axes. 

Show that the general conditions of the circle render tan 2a inde- 
terminate. 

Prove, by the last value of tan 2a and the formulae for the 
double angle, that when B = 2C cos w, the axis of Y is parallel to 
an axis of the curve. 

317. To find the semi-axes, it is necessary to reduce the equation 
to the central form, Ax 2 -|- 3xy -\- Cy 2 -j- F' = ; and then, making 
the axes rectangular, to reduce it to the form, 

av -f cy + F' = 0. 

Whether the axes are rectangular or oblique, F' is computed by 
the formula of Art. 289. Now supposing the axes rectangular, the 
values of A' and C may be found by means of the quantities, 
proved in Art. 314 to be constant in value. For since B' = 0, 

A' + C' = A+C and 4A'C = 4AC — B 2 . 

We have therefore two equations by which to find the values of A' 
and C'i The form of these equations is such as to give values of 
A' and C, which may be interchanged ; the reason of which is that 
the axis of X may be made to coincide with either axis of the 
curve. 

The equations may be solved thus : Squaring the first* and sub- 
tracting the second member from member, we have (A' — C') 2 = 
(A — C) 2 + B 2 , hence A' — C .= ±l/(A — C) 2 + B 2 . Since the 
quantity under the radical sign is essentially positive, the values 
of A' and C are always possible. To determine which sign to give 
to A' — C, observe that, by the values in Art. 314, A' — C = 
2B sin a cos a ; hence if we take a in the first quadrant we must give 



RECTANGULAR EQUATIONS. 231 

to A' — C the sign of B. From the values of A' — (7 and 
A' -|- C we readily find those of A' and C. The squares of the 
semi-axes are numerically equal to the squares of the intercepts 
in the reduced equation ) the intercepts themselves being both real 
for the ellipse, one real for the hyperbola, but neither real for the 
imaginary ellipse. 

Examples. — The axes being rectangular, transform each of the 
following conies to its axes ; x 2 — 4xy — 2y 2 = 36 ; 5x 2 -|- 3xy -j- 
f = 1 ; Qx 2 — bxy — Qy 2 = — 26. 

Determine that semi-axis of 2x 2 -j- 4xy -f- 5y 2 -f- Qx — 2y = 
which makes an acute angle with the axis of X. 

318. In order to find the values of A' and C directly from 
the oblique equation, let us find the value of the constant quantity 
A -j- C (in the rectangular equation) in terms of the general or 
oblique coefficients. Adding the values of A' and C, the rectangu- 
lar coefficients of Art. 312, we find 

A , + c , = A + C - - B cos to 



sin to 

Now, by Art. 314, A' -j- C has a constant value independent of 
the direction of the axis of X; therefore the expression in the 
second member, which involves only the mutual inclination of the 
axes, is constant in value. By this, and Art. 292, we see that 

A -f C — B cos a, _ 4AC — B 2 
! and 



sin^ w sin to 

are two functions of the coefficients and the angle between the co-ordi- 
nate axes, of which the values are unchanged by transformation. 
When (o = 90°, these expressions reduce to those given in the last 
Article; and in all cases, they give us the values of A' -j- C and 
4A'C by which A' and C may be found. 

Examples. — The inclination of the axes of co-ordinates being 
60°, give the equation of each of the following conies as referred 
to its axes; 3x 2 -f- 3xy -f 4y 2 — 12 = 0; x 2 — xy — y 2 = \) 2x 2 + 
^ + % 2 = 2 ; x 2 — xy-\-y 2 = 4. 

Transform to its axes each of the following, supposing w = 120° : 
x 2 — xy -j- y 2 = 1 ; 4x 2 — 4:xy -f 3/ — 2 = ; xy — y 2 = 1 • 
2x 2 — xy -}- y 2 = 1. 

Transform to its axes 3x 2 — 2xy — 3y 2 = 6, when cos to = £. 



232 GENERAL EQUATION OF THE SECOND DEGREE. 

Transform to its axes Ax 2 -j- Bxy -J- A.y* == 1. 



* 2 + , — — y=i 



1 -j- cos w 1 — cos 

The equation of the ellipse referred to conjugate diameters is 

x 2 ip' 

— -\- — = 1. If it be transformed to any other pair of conjugate 
a 2 6 2 

diameters it will retain the same form, but a and b will take new 
values. Prove that a 2 -f- b 2 and ai sin io are constant. (Substitute 
the coefficients of x 2 and ^ 2 for A and C in the above expressions.) 
Prove, in a similar manner, that the sum of the squares of the 
reciprocals of perpendicular semi-diameters is constant. (Assume 
the rectangular equation Ax 2 -J- Bxy -f- Qy 2 -j- F = 0, and take the 
intercepts on the axes for the semi-diameters.) 

Conic Fulfilling Given Conditions. 

319. Since the general equation of the conic section contains six 
coefficients, whose ratios determine the position and shape of the 
curve, it may be regarded as containing five arbitrary constants, 
which may be so determined as to make the conic fulfil five given 
conditions. Thus, a conic passing through five given points may 
be found ; for, assuming a value for the absolute term, the values of 
the five coefficients may be determined by five equations of con- 
dition expressing that the curve shall pass through the five given 
points. Since these equations are of the first degree, they have 
but one solution ; therefore one conic, and generally but one, can 
be found fulfilling the conditions. If three of the given points are 
in the same straight line, the conic which we shall find will consist 
of this straight line and that which passes through the two other 
points. If four of the points are in one straight line, the conic will 
be indeterminate, because this straight line in connection with any 
straight line passing through the fifth point will fulfil the conditions. 

320. In general a conic can be made to fulfil any five conditions. 
That it shall be a parabola constitutes one condition, because this 
gives the equation B 2 — 4AC = between the coefficients. That 
it shall be an ellipse is only a restriction, because this does not give 
an equation between the coefficients, but only requires that 4 AC shall 
exceed B 2 . A parabola may generally be found passing through four 



CONIC FULFILLING GIVEN CONDITIONS. 233 

given points, because four equations of condition together with 
B 2 == 4AC serve to determine the five unknown quantities, or 
ratios of the coefficients. But, as this last equation is of the second 
degree, there may be two solutions. In fact, it is evident that two 
parabolas may intersect in four points. The solution may be im- 
possible, because the results may be imaginary quantities. This will 
happen if the four points are so situated that one may be enclosed 
in the triangle formed by joining the others, for it is evident that 
four points of this character cannot be found on a parabola. 

To be similar and parallel to a given conic, or to have given 
directions for the asymptotes, constitutes two conditions, because it 
determines the ratios of the coefficients of the three terms of the 
second degree, upon which, as we have seen, depend the directions 
of the asymptotes, conjugate diameters and axes. The parabola 
having a given direction for its diameters and axis is a case of this. 
If the axis is to be parallel to y = mx -\- b, the equation may be 
assumed in the form 

(y — mx) 2 -f J)x -f Ey -f F = 0, 

in which we have assumed C = 1 (see Art. 286), so that the re- 
maining coefficients may be determined by three other conditions. 

To be a circle also constitutes two conditions, for the conditions 
of Art. 313 determine the ratios of A, B and C. If we wish to 
find the circle passing through three given points, we may assume 
the equation in the form 

x 2 -f- 2xy cos w -L y 2 -f Dx -j- Ey -j- F = 0. 

Examples. — Determine the parabola whose axis is parallel to 
y = 2x, and which passes through (1, — 1) ( — 2, 0) and ( — 1, 2). 

If cos id = j, what is the equation of the circle passing through 
these points? 

321. A pair of straight lines constitutes a conic fulfilling one 
condition, for the coefficients must satisfy the equation of Art. 289. 
If we attempt to determine the conic by means of this equation and 
four equations of condition expressing that the curve shall pass 
through four given points, the result of elimination will be an equa- 
tion of the third degree containing one unknown quantity. There- 
fore there may be three solutions. The interpretation of this is that 
20* 



234 GENERAL EQUATION OF THE SECOND DEGREE. 

four given points may be joined by pairs in three different ways, 
giving three pairs of straight lines fulfilling the conditions. The 
three equations of the second degree may be found by compound- 
ing the equations of the straight lines, as in Art. 81. 

A pair of parallel lines is a conic fulfilling two conditions; 
namely, the condition for straight lines and that of the parabola. 
A pair of coincident lines fulfils a third condition, so that it can be 
made to satisfy only two more conditions, or to pass through two 
given points. 

If S = represent the equation of a conic, and a = 0, /9 = 0, 
the equations of straight lines, then S is a polynomial of the second 
degree involving x and y, while a and /9 are of the first degree. 
Then the condition for straight lines is equivalent to this : that the 
polynomial S shall be the product of two factors of the first degree, 
or that the equation shall be capable of taking the form a/9 == 0. We 
here use the Greek letters to denote expressions of the form 
Ax -j- By -j- C, then, since a = contains two arbitrary constants 
and fi =*= contains two, a/9 — contains four. 

The condition for coincident lines is that the equation shall be 
reducible to the form a 2 = ) in other words, that the polynomial S 
shall be the square of an expression of the first degree. 

322. If S = and S' = are the equations of two given conies, 

S + ifeS' = 
is the equation of a conic passing through their points of inter- 
section. In Art. 270, we supposed the equations combined to 
be of the form in which B = j that is, the term containing 
xy did not appear in the equations. Therefore the given conies 
already fulfilled one condition, and the whole system of conies which 
may be represented by S -j- kS' = fulfilled the same condition, as 
explained in Art. 271. We saw, in that case, that the conies may 
intersect in four points j but we could not use the formula to pro- 
duce the equation of the conic passing through four given points, 
because it is generally impossible to find more than one conic of 
that form which shall pass through four given points.* Using now 

* See Note to Art. 271, in which it was shown that three points and the 
condition implied in the form of the equation determine the fourth point, 
so that S -f- &S' =0 does not fulfil five independent conditions. Four points 



CONIC FULFILLING GIVEN CONDITIONS. 235 

the general form of the equation, this may readily be done by taking 
the equations of the pairs of lines mentioned in the last Article. 
Thus, given four points A, B, C and D, form the compound equa- 
tion of the lines AB and CD, also that of the lines AC and BD„ 
These constitute the equations of two conies passing through the 
four points : they may, therefore, be used in forming the general 
equation S -\- &S' = 0. It is now possible so to determine k, as to 
make the conic pass through a fifth given point. 

323. This method may be adapted to other cases of conies ful- 
filling four conditions. Thus, to be tangent to a given line at a 
given point constitutes two conditions, for it implies that the conic 
shall there intersect the given lines in two coincident points. If 
now we require the equation of a conic touching a given line at 
the point A and passing through B and C, we have only to take 
for S = 0, the compound equation of the given line and BC, and 
for S' = that of the pair of lines AB and AC. For it is 
evident that the later conic intersects the given line only at the 
point A, and the conic S -j- &S' == cannot meet it in any other 
point, because it cannot meet the first conic in a point not on 
the second. For instance, let the conic be required to touch 
2x — y -f- 1 = at (1, 3) and to pass through ( — 1, 2) and the 
origin. The first conic is (2x — y -j-. 1) (2x -f-y) = or 4x 2 — 
y 2 -j- 2x -f- y = 0, and the second is (x — 2y -f- 5) (3a; — y) = 
or 3x 2 — 7xy -f- 2y 2 -j- 15sc — 5y == 0. Combining the equations, 
we have 

4x 2 — y 2 -)- 2x -f y -f h (3x 2 — 7xy -f 2y 2 -f 15a: — 5y) = 0, 
in which h may be determined by another condition. 

generally determine the direction of a pair of conjugate diameters. For, 
suppose two parabolas to pass through them. Take two straight lines, of 
which each is parallel to the axis of one of these parabolas. The equa- 
tions of the parabolas, referred to these lines as co-ordinates axes, will be, 
one of the form Ax 2 + Da; + Ey + F = 0, the other of -the form C^ 2 + 
D'x -j- Wy + F r = 0. Denoting these equations by S = and W = 0, we see 
that S -j- A;S / =0 will always be of the form in which B — ; therefore every 
conic passing through the four points has a pair of conjugate diameters 
parallel to the axes of the parabolas. When the lines joining the given 
points form a quadrilateral with a re-entrant angle, hyperbolas only can be 
drawn through them, and there are no parallel pairs of conjugate diameters. 



236 GENERAL EQUATION OF THE SECOND DEGREE. 

324. For the equation of the conic tangent to two given lines at 
given points, take for S =± the pair of given lines, and for $' = 
the pair of coincident lines passing through the given points. Thus, 
let the conic be required to touch the axis of X at (3, 0) and the 
\me*x=y at (1, 1). The first conic is y (x — y) = 0, and the 
second is (x -\- 2y — 3) 2 = 0. Hence the required equation is 
x 2 -j- 4xy -j- 4j/ 2 — 6x — 12y — | — — ( — kxy — ky 2 = 0, in which h 
may be determined by another condition. For instance, required 
also that it be a parabola. In the equation, A = 1, B =4 -j- &, 
C =? 4 — k, therefore B 2 = 4AC gives (4 -f- k) 2 = 4 (4 — k\ 
which reduces to 12k -\- k 2 = 0. This is satisfied by k = 0, be- 
cause the equation will then reduce to that of the coincident lines 
which are of the form of the parabola ; the other solution is 
k = — 12, which gives the true parabola x 2 — 8xy -\~ lQy 2 — 
6x — 12y -f 9 = 0. 

In a similar manner we can find the equation of a conic meeting 
a given conic in four points of which one pair or two pair may be 
coincident. It is also possible to make three of the points of inter- 
section coincide ; for combine with the given equation that of a 
pair of lines, one of which is a tangent and the other passes through 
the point of contact. The conic S -j- kBi = is in this case said 
to make double contact with S = 0. By taking forS' = the 
square of the equation of a tangent, S -f- k& = may be made to 
meet S = in four coincident points, in which case it is said to 
make a contact of the third order. 

325. Any equation of the second degree, in which one constant 
is arbitrary, may be regarded as an equation of the form S -j- k$ r = 0, 
in which k represents the arbitrary constant, and S and S' are 
polynomials, of which at least one is of the second degree. Though 
g = and S' = may not actually intersect in four points, 
S -f- &S' == always fulfils four conditions. Thus, if S' is of the 
first degree, S -{- kS' = will be parallel and similar to S = 0, and 
will intersect it in the two points where it is cut by the straight 
line S' = 0. The equation of the parabola with axis parallel to a 
given line, and passing through two given points, may be formed, by 
taking for S = the pair of lines parallel to the given lines and pass- 
ing through the given points, and for S' = the single straight line 
passing through the points. That of the hyperbola having asymptotes 



CONIC FULFILLING GIVEN CONDITIONS. 237 

in given directions may be found, by taking for S = the compound 
equation of two straight lines, each of which passes through one 
of the given points and has one of the given directions. 

326. The general equation Ax 2 -f Bxy -f- Cy 2 -f T>x -j- Ey + 
F t= may itself be regarded as one of the conies represented by 
S -f JcS' = 0, when S = is the pair of straight lines passing 
through the origin, Ax 2 -f Bxy -f Cy 2 = 0, Art. 286, and S' = 
is the single line Dx -J- Ey -\- F = 0. From this we might at 
once infer that the lines Ax 2 -f- Bxy -j- Cy 2 = meet the curve 
each in a single point, and that the terms of the first degree 

D:e_|_Ey-r-F = 0, 

constitute the equation of a straight line passing through the points 
in which these lines meet the curve. Since the pair of lines 
Ax 2 -j- Bxy -j- Cy 2 = is real for the hyperbola, this line cuts the 
curve in the points where it is cut by two lines passing through the 
origin parallel to the asymptotes. For the parabola, it is a tangent 
to the curve at the point where a line passing through the origin 
parallel to the axis cuts the curve. In the case of the ellipse, it 
does not meet the curve. 

The four conditions fulfilled by S -j- &S' = may always be 
considered as determining four common points of intersection for 
the whole series. But these points may be real, coincident or im- 
aginary, as explained in Art. 272; and in the special cases of 
similar ellipses and hyperbolas in which one or both pairs of asymp- 
totes are parallel or coincide, some of the intersections become infi- 
nite. Compare Art. 273. 

Examples. — Find the equation of the conic passing through 
(3, 1), (2,-2), (— 1, — 1), (0, 2) and (2, — 3) by the method 
of Art. 322. 

Give the general equation of the conic tangent to y — 2x — 3 
at (2, 1) and passing through (1, — 2) and (3, 1) j and determine 
h so as to make the conic pass through the origin. ' 

Give the general equation of the conic touching x 2 — xy -\- 
4cy 2 -j- 2x — 3y -f- 4 == 0, at the points where it is cut by the 
straight line 2x — y -{-1 = 0. What must be the inclination of 
the co-ordinate axes in order that a circle may be found fulfilling 
these conditions ? 



238 GENERAL EQUATION OF THE SECOND DEGREE. 

Determine the parabola parallel to 3x -J- 2y = and passing 
through (1, 1), (2, — 2) and the origin. 

If cos <o — i, what is the equation of the circle tangent to 
y — x at the origin and passing through ( — 2, 1) ? 

What is represented by the general equation, supposing the co- 
efficient A to be arbitrary ? 

Ans. A series of conies having common tangents at the points 
where they cut the axis of Y ; for x 2 = is the equation of two 
lines coincident with that axis. The equation of the pair of tan- 
gents may be found by determining A as in the first example 
under Art. 289. 

Interpret the equation in a similar manner, regarding each of 
the coefficients in turn as the arbitrary constant. 

Supposing a = 0, ft = 0, etc., to represent straight lines, what is 
denoted by aft -f kyd = ? by ap+kf — 0? by a 2 -j- kft 2 = ? 
by S + £a 2 =0? by aft-\~ky=0? by aft + ka = 0? by a 2 + 
kft = Q? by S + A£ = ? 

What is denoted by aft -f F = ? 
Ans. An hyperbola whose asymptotes are a = and § = 0. 

If S = and S' = have a common asymptote, show that 
S -f- &S' == has the same line for one asymptote. (If a = is the 
equation of the common asymptote, S = and S' = may be writ- 
ten in the forms aft -j- F = and ay + F = 0; and S+ M = 
is equivalent to a (ft -f ky) -J- F -j- && = 0, an hyperbola whose 
asymptotes are a = and ft -\- ky= 0. That is, S -f- kS = 
represents a series of hyperbolas of which a == is a common asymp- 
tote, and the other asymptotes pass through a common point, the 
intersection of ft = and ^ = 0.) 

Equation of a Polar. 

327. The general formula for the tangent at a given point P x 
found in Art. 308, is 

(2A^ + By, + D) (x - x x ) + (2%, -f Bx x + E) (y — y x ) = 0. 

This is always the equation of a line passing through P 1} and it 
represents a tangent when 

Kx 2 + Bx x y x + (V + Dx x + %, + F -r ; 



EQUATION OF A POLAK. 239 

that is, when P x is a point of the curve. * If we partially expand 
the equation of the tangent and add to it twice this equation of 
condition, it will take the form 

' (2M -f By, + D)x + (20^ + Bx, + E)y + Dayf Ey x + 2F == 0. 

Now this is the equation of the polar of P x ; for dividing by 2 and 
arranging the terms in the order of the coefficients, we have 

A^-j- *B (xy x +yx,) + Cyy x + *D (*+ aQ+fE (y+ yi ) + F== 0. 

When written in this form, it is evident that if P 2 is on this line, it 
is connected with P x by a reciprocal relation. Such points, there- 
fore, may be defined as reciprocally polar, and the straight line as 
the polar of P v It will be seen that this is the most general 
formula connecting points by a reciprocal relation of the first degree 
with respect to the co-ordinajtes of each. The condition that a point 
shall be on its own polar is the same as that which expresses that 
it is on a certain conic, and the relations between the curve, pole 
and polar, given in Art. 277, are of general application, because 
they are independent of the co-ordinate axes. 

328. If it be required to find the equations of tangents passing 
through a given point, the co-ordinates of the points of tangency 
may be found by combining the equations of the curve and the 
polar of the given point. 

The polar of the origin is found by the formula to be 

Dx + Ey -f 2F = 0. 

Comparing this with Dx -j- Ey -f- F = 0, which by Art. 326 passes 
through the points where the curve is cut by parallels to the asymp- 
totes drawn through the origin, we see that it is parallel to that line 
and twice as far from the origin. As any point may be taken as 
origin, we derive the following property of the hyperbola : If from, 
a given point tangents be drawn and also lines parallel to the asymp- 
totes, the line joining the points of section will be parallel to that 
joining the points of tangency and midway between it and the given 
point. 



CHAPTER IX. 

GEOMETRICAL LOCI. 

329. In this Chapter, the principles of Analytical Geometry are 
applied to the problem of finding the locus, or path, of a point 
moving according to a given geometrical law. The method con- 
sists in assuming co-ordinate axes, to which to refer the moving 
point; establishing a relation between its co-ordinates, equivalent 
to the law of the point's motion ; and finally, interpreting the rela- 
tion found, which is the equation of the locus, so as to ascertain its 
character and position. 

The law by which the describing point moves may be stated in 
the form of a condition imposed upon the point, suflicient to restrict 
it to a certain line, but not to determine its position. Hence a locus 
corresponds to an " indeterminate equation," or single equation be- 
tween x and y, the unknown co-ordinates of a point. See Arts. 15 
and 16. We may, also, regard the law of the motion as the state- 
ment of a common property of all the points of the line sought. 
From this property we are to deduce the equation of the line, which 
also expresses a common property of all its points. If the equation 
found is included under any of the general equations which we 
have investigated — that is, if it is of the first or second degree — the 
form and position of the line having the given property become 
known. Thus, in the preliminary illustration of Art. 22, the re- 
sult being the equation of a circle, we learn that it is a property of 
the circle that the squares of the distances of any point of the cir- 
cumference from certain fixed points have a constant sum. 

Choice of Co-ordinate Axes. 

330. When the problem is simply to find the line described by 
a point, and its position with reference to certain fixed lines and 

240 




CHOICE OF CO-ORDINATE AXES. 241 

points, the readiness with which we can establish the equation be- 
tween x and y depends upon the manner in which we assume the 
axes. We select as an illustration the following problem : 

Given two fixed intersecting lines and a fixed point. A, a line is 
drawn through A, meeting the fixed lines in B and C ; find the locus 
of the middle point o/"BC. 

Take the two fixed lines, OX and OY, as axes, and draw the 
ordinate of the fixed point A, and 

that of the middle point P whose -. c, 

locus is required. Denote the co- 
ordinates of A, which are constant, 
by a and b; those of P are of course 
x and y. We have now to establish 
a relation between x, y, a and &, by 

means of the conditions of the problem and the geometrical princi- 
ple of similar triangles. Since P is the middle point of BC, OB = 2x 
and OC = 2y j and by similar triangles 2y : 2x : : b : 2x — a ; hence 

bx = 2xy — ay. 

This is the equation of the locus required, which is therefore an 
hyperbola with asymptotes parallel to the fixed lines OX and OY. 
The equation also shows that the locus passes through the fixed 
point A (a, 5) \ and through the intersection of the fixed lines. By 
Art. 282, the centre of the hyperbola is the point (£a, W) midway 
between and A. 

331. It is evident, on the first statement of this problem, that 
the conditions are not sufficient to fix the point P, because the line 
BC is not fixed. But they limit the position of P to a locus, which 
is described by the point as the line BC revolves about the point 
A. Accordingly, we find it possible to establish a single equation 
between the co-ordinates of P, which leaves its position indeter- 
minate, though restricted. If the conditions had been sufficient 
to fix the point P, we should have been able to establish two 
equations between x and y, and their values would have been deter- 
minate. 

In solving problems of this kind, it is necessary to distinguish 
carefully between the constant and variable lines in the figures. 
To do this we must consider the motion which takes place among 
21 L 



242 



GEOMETRICAL LOCI. 



the parts of the figure, as, in the above example, the revolution of 
the line BC, by reason of which OC and OB are variables. The 
constant distances of the figure should be denoted by letters, so that 
the solution may be general j the discussion of the problem then 
consists in examining the special cases which may occur, when the 
parts of the figure have particular relative positions. Thus, in the 
present case, if the fixed point A be on one of the fixed lines as 
OX, so that b = 0, the equation reduces to 2xy — ay = 0, which 
represents the two straight lines y — and 2x = a. Therefore in 
this case the locus becomes two straight lines, one of which is 
parallel to OY and bisects the distance OA, and the other is the line 
OX itself. 

If the complete revolution of BC about the point A be con- 
sidered, it will be seen in what manner the two branches of the 
curve are described by P, in the general case. In the special 
case, it will be observed that the line OX is not, strictly speaking, 
described by the motion of P; but it constitutes part of the locus, 
because the position of B, which, in this case, generally coincides 
with A, becomes indeterminate when the revolving line coincides 
with OX, and P is then an indeterminate point of that line. 

332. To find the locus of a point moving in such a manner that 
the square of its distance from a fixed point is proportional to its 
distance from a fixed line. 

Take the fixed line as the axis of X, and since the condition of 
the problem involves distances, let the axes be rectangular; the 
axis of Y may be taken so as to pass 
through the fixed point. Then will 
PR, the distance from the line, be de- 
noted by y. Let b denote the distance 
OB of the fixed point from the fixed 



: PR : : c : 1. 

b) 2 , therefore — 



line, and assume PB 2 
Now PB 2 = x 2 -f (y — 
* 2 ~f O — bf = cy, or 



ar 2 -f y 2 — (2b -f c) y -f b 2 = 0. 

Hence the locus of P is a circle whose centre is on the line OB, 
and which generally does not cut the axis of X. But in the special 
case when the fixed point is on the fixed line ; that is, when b = 0, 



APPLICATION OF ANALYTICAL FORMULA. 243 

the equation becomes x 2 -j- y 2 — cy = 0, which represents a circle 
having for diameter c, and touching the axis of X or given line at 
the origin or given point. 

When the conditions of a problem make two variable quantities 
proportional, we may always assume one of them equal to the other 
multiplied by a constant. Thus, in this example, PB 2 = cPR, in 
which c represents a constant third proportional to PR and PB. 

Examples. — A line of fixed length moves with one of its ex- 
tremities in each of two fixed lines : find the locus of any point of 
the line. (Let a and b be the distances of the point from the ex- 
tremities of the line, and discuss the cases a = and a = 6.) 

A line cuts two fixed lines, OX and OY, in B and C, and moves 
in such a manner that the area of the triangle OBC is constant ; 
find the locus of the middle point of BC. 

Find the locus of the middle point of a rectangle inscribed in a 
given triangle. (Assume the base of the triangle as the axis of X, 
and a perpendicular through the vertex as axis of Y, so that the 
axes shall be parallel to the sides of the rectangle.) 

Given the base and sum of the sides of a triangle, a perpendicu- 
lar to the base is drawn through the vertex and produced to equal 
one of the sides, find the locus of its extremity. 

Aiis. A straight line. 

A point moves so that the squares of its distances from two fixed 
points are as m : n, what is the locus described ? 

A line is drawn through a fixed point A, cutting a fixed line in 
D ; through the point R of the fixed line is drawn a line PR cut- 
ting AD in P ; find the locus of P, supposing DR to be constant, 
and the line PR to cut the fixed line at a constant angle. (Take 
the fixed line as axis of X, and for axis of Y, a line passing through 
A parallel to PR.) 

Application of Analytical Formula. 

333. In the foregoing examples, we have been able to establish 
the relation between x and y by means of simple geometrical prin- 
ciples. It is frequently necessary to employ the principles of anal- 
ysis for this purpose, especially when the axes are already deter- 
mined. For example, let it be required to find the locus of a point, 
when the square of its distance from a given point P x is proportional 



244 GEOMETRICAL LOCI. 

to its distance from the line x cos a -J- y sin a =p, the axes being 
rectangular. Using the formulae for the distance of points, and 
the distance of a point from a line, the condition of the problem 
gives the equation 

(x — x x ) 2 -j- (y — y x y = c (x cos a -f- y sin a — jp), 

which when expanded evidently represents a circle. When P x is on 
the given line it may be shown that the equation represents a circle 
touching the given line at P 1? and having c for its diameter, as in 
the special case discussed in the last Article. It will be seen that 
the result of this method is a general formula, but that the mere 
solution and discussion of the problem is more simple when we are 
at liberty to assume the axes. 

334. When a variety of points is given, the most symmetrical 
and useful solutions will be obtained by using general expressions 
for all of them. Thus : To find the locus of P, when the sum of the 
squares of its distances from any number of fixed points is constant. 

Let P l5 P 2 P n be the points (n being the number of points), 

and suppose the axes rectangular; then, denoting the constant sum 
by c 2 , we have 

(x— x l f+(x-x 2 y.. + (x-x n y+(y-y i y.. + (y-y n y = c>. 

Expanding and dividing by n, we find this to be the equation of a 
circle, the co-ordinates of whose centre are arithmetical means be- 
tween the corresponding co-ordinates of the given points. 

Examples. — Find the locus of a point whose distance from a 
fixed point is proportional to its distance from a fixed line. (Take 
the fixed line as the axis of Y, and assume the first distance equal 
e times the second. The result will be a conic section of which e 
is the eccentricity, the fixed point the focus and the fixed line the 
directrix.) 

Find a formula for a conic having the point P x for focus, and 
the line x cos a -\- y sin a =%>, for directrix. 

Show that the locus of a point, the square of whose distance 
from one fixed line is proportional to its distance from another, is a 
parabola whose axis is parallel to the first line. (Assume rectan- 
gular axes and find a general formula ; also assume the first line as 
axis of X and prove that the second line is a tangent, by Art. 326.) 



ELIMINATION OF VARIABLES. 



245 



Elimination of Variables. 

335. In many cases it is convenient to use other variables besides 
x and y, and to derive from the conditions a sufficient number of 
equations to eliminate these auxiliary variables, so as finally to have 
a single equation between x and y. The methods we used in find- 
ing the equations of the ellipse and hyperbola were instances of 
this : three equations were found between the four variables, 
x, y, r and /, and were reduced to a single equation between two 
variables, just as in Algebra any number of equations containing 
an equal number of unknown quantities is reduced to a single 
equation containing one unknown quantity. The number of equa- 
tions must be one less than the whole number of variables, in order 
that the position of P may be indeterminate. 

As an illustration, we solve the following problem : Find the locus 
of the point, in which the perpendicular from the centre of an 
upon a tangent cuts the ordinate of the point of contact. 

Let P be the point of inter- 
section, and let <p be the eccen- 
tric angle of the point of con- 
tact, P x . Then the equation 
of the line CP, perpendicular 
to the tangent (or parallel to the 
normal, Art. 200), and passing 
through the centre is 

B cos <p . y = A sin <p . x. 

Now the abscissa of P is the same as that of P 1? therefore 

A 2 
x — A cos cp ) and since P is a point of the above line y = — sin ^, 




B 



hence 



cos <p = 



and 



sin tp = 



By 



Eliminating <p by substitution in sin 2 (p -j- cos 2 <p = 1 , we find 

By 



x 
A 2 



= 1, 



21 



246 



GEOMETRICAL LOCI. 



the equation of an ellipse, whose semi-axes are A and 



B 



The 



latter being a third proportional to B and A, the locus is similar 
to the given ellipse, and its minor axis coincides with the given 
major axis. 

336. When the motion which takes place in describing the locus 
involves a constant change in the direction of certain lines, it will 
often be necessary to use as an auxiliary variable an angle dependent 
upon their directions. Thus : 

A given triangle moves with two of its vertices in the two rectan- 
gular axes of co-ordinates ; find the locus of the third vertex. 

Let PAB be the given triangle ; drop a perpendicular from the 
origin upon the base AB, and let 0, denoting the inclination of 
this perpendicular, be the auxiliary 
variable. Since PAB is a given tri- 
angle, any of its parts may be used as 
constants. It will be most convenient 
to use the perpendicular from P, and 
the segments into which it divides the 
base, denoted by p, a and 6, as in the 
figure. These constants and the value 
of 0, at any stage of the motion, de- 
termine the position of P. Dropping 
extremities of p, we easily show that 




perpendiculars from both 



x =p cos -j- o sin 



and 



y =p sin -f- a cos 0. 



These are the two relations between x, y and 0. We eliminate by 
finding values for sin and cos and using the relation sin 2 -j- 
cos 2 0r=l. Thus, eliminating successively sin and cos 0, we 
have 

(p 2 — ah) cos ==px — by and (p 2 — ah) sin =py — ax. 
Squaring and adding member to member, 

(p 2 — ab) 2 = O 2 4- a 2 ) x 2 -f (p 2 + b 2 )y 2 — 2p (a -f b) xy. 
This is the required equation between x and y : by the principles 



ELIMINATION OF VARIABLES. 247 

of the last Chapter, it represents an ellipse, with centre at the 
origin, but not having the lines OA and OB for axes.* 

Examples. — Find the locus of the point in which a parallel to 
the axis of X drawn through P x (Fig. Art. 335), cuts the line PC. 

Given two fixed straight lines intersecting in 0, a straight line 
cuts them in A and B, and P is such a point of the line that 
AP : BP : : ra : ». Find the locus of P, 1st, when OA + OB is 
constant j 2d, when A 2 -j- OB 2 is constant ; 3d, when A X OB is 
constant. (Let v and z, denoting respectively OA and OB, be the 
auxiliary variables. By similar triangles their values are readily 
expressed in terms of x and y.) 

Find the locus of P when AB constantly passes through a fixed 
point. (A relation between v and z may be found by expressing 
the condition that the fixed point (a, 6) is on the line whose inter- 
cepts are v and z.) 

A straight line passing through a fixed point cuts a given conic, find 
the locus of the middle point of the chord. Take the fixed point as 
origin. Then we may assume Ax 2 -f- Bxy -j- Q>y 2 -}- Dx -{- Ey -f~ 
F = for the equation of the conic, and y = rax for the straight 
line. The abscissas of the points of intersection are the roots of 
(A -f raB + ra 2 C) x 2 -f (D -f- raE)x -f- F = 0. It is not neces- 
sary to find these roots, for their sum is the negative of the coeffi- 
cient of x divided by that of x 2 ; hence the abscissa of the middle 
point is 

D -f raE 

X = J , 

A + mB + ra 2 C 

y 
but ra = -, because the point is on the line y=. rax. Substitut- 

x 

ing, etc., we have 

* The equation of this locus will take a simpler form if we introduce 
other constants in place of a, b and p. Thus, let c denote the side PA, d 
the side PB and a the included angle at P. Then p 2 -f- a 2 =c 2 ,p 2 -{- b 2 =d 2 , 
p (a -f- b) = cd sin a (twice the area of the triangle), and p 2 — ab = 
i [c 2 + d 2 — (a-f b) 2 ~\ =cdcosa (since by Trigonometry (a -f- b) 2 =c 2 + 
d 2 — led cos a), hence the equation becomes 

c 2 x 2 + d 2 y 2 — 2cd sin a .xy = c 2 d 2 cos 2 a. 

The reader mav discuss the cases a = 0°, a = 180° and a = 90°. 



248 GEOMETRICAL LOCI. 

Ax 2 + Bxy + Cf + %I)x + £Ey = 0. 

Tlie required locus is therefore a. similar conic passing through the 
fixed point, and by Art. 287, its centre is midway between that 
point and the centre of the given conic. 

Intersection of Variable Lines. 

337. The variable which we have to eliminate may be, as in 
some of the examples already given, one of the quantities which 
determine the position of a line whose equation we employ. Thus, 
in Art. 335, <p appears in the equation of CP. We have hitherto 
called such quantities constants, in distinction from x and y, because 
their values are determined by the position of & fixed line. When 
the line moves according to a certain law, these determining quan- 
tities become variable, as when we discuss the equation of a line, 
considering one of its constants arbitrary. 

If now the locus required is described by the intersection of two 
moving or variable lines, it is necessary to express the equation of 
each of these lines in terms of constants and a single auxiliary vari- 
able. Then we shall have two equations between x, y and the 
auxiliary : the latter may therefore be eliminated. Thus : 

A line of given length moves with one of its points in a fixed line 
with which it makes a constant angle ; its extremities are joined by 
straight lines to two fixed points of the fixed line ; find the locus of 
the intersection of the joining lines. 

Take the fixed line as axis of X, one of the fixed points as origin, 
and the axis of Y parallel to the line of given length. Let b and c 
be the constant parts of this line above and below the axis, and a 
the distance OA between the 
fixed points. Let z represent 
the variable abscissa of D. 
Then the equation of OB 
passing through the origin 
and B (z, 6) is yz = bx, and 
that of CA passing through 

(a, 0) and (2, — c) is y = — (x — a), or yz — ay = — ex -J- ac. 

z — a 

Since P is on each of these lines it satisfies both their equations, 




INTERSECTION OF VARIABLE LINES. 249 

therefore we have two relations between the co-ordinates of P and 
the auxiliary variable z. Eliminating the latter by subtraction, 

ay = (b -f- c ) x — ac - 

The locus is therefore a straight line, passing through the points 

(0, — c) and (a, 6). 

338. Instead of a single auxiliary, in expressing the equations of 

the variable lines, it is sometimes convenient to use two, between 

which the conditions of the problem give a relation. Thus, if we 

require the locus of the intersection of lines passing through the 

points (a, 0) and ( — a, 0), the equations are y =m (x — a) and 

y = m! (x -f- «)• Then, if the conditions give an equation between 

y y 

m and ra', we may substitute the values m— , m = , 

x — a x -j- a 

and the result will be the required locus. Supposing the axes rec- 
tangular, if the lines are to be perpendicular, the equation connect- 
ing m and m! is mm! = — 1, and the locus will be found to be a 
circle. In general, if mm! is constant, the locus is a central conic 
of which 2a is an axis. If m -{- m! is constant, the locus is always 
an hyperbola, and if m — m is constant, it is a parabola.* 

Examples. — Find the locus of the intersection of tangents to 
an ellipse at the extremities of conjugate diameters. (Since by 
Art. 195, the eccentric angles of the vertices differ by 90°, use the 
equations of the tangents in terms of the eccentric angles (p and 
<p -f 90°, Art 192. Eliminate <p as in Art, 336.) 

Find the locus of the intersections of perpendicular tangents of 
an ellipse. (We may use the form of Art. 190, or that of Art. 193; 
the locus will be found to be a circle as proved in Art. 194.) 

Show that the locus of the intersection of a tangent to a conic, 
with a perpendicular from a focus, is a circle ; but that of the inter- 
section with a perpendicular from the centre is of the fourth degree, 
except when the conic is a circle. 

A line parallel to the side BC of the triangle ABC, cuts AB in 
D, and AC in E, find the locus of the intersection of BE and CD. 

* Since m and m' are the tangents of the inclinations of the lines (axes 
being rectangular), we may find the loci when the sum or difference of the 
inclinations is constant, or when one is double the other, by the angular 
analysis.— See Chauvenet's Plane Trig., Eqs. 123, 124, 137. 

L* 



250 GEOMETRICAL LOCI. 



Point Connected with a Variable Line. 

339. The equation of a variable line either contains one arbitrary 
constant, or constants connected together by such relations that all 
the rest may be expressed in terms of any one of them. For in- 
stance, the position of the straight line y = mx -j- b is determined 
by two quantities m and b ; when they are both regarded as arbi- 
trary, the line is wholly indeterminate, but when they are connected 
by an equation, such as b 2 == A 2 m 2 -j- B 2 (which is the condition 
of tangency to the ellipse, Art. 190), the line is conditioned, but not 
determined* Other lines require more than two determining quan- 
tities, and are therefore indeterminate whenever these quantities 
are connected by a number of equations less than their own num- 
ber ; but the variable lines here considered are those in which the 
number of equations or conditions is one less than the number of 
quantities, so that one of them may be regarded as a variable of 
which all the rest are functions. 

340. Now suppose that there is a point connected with a variable 
line ; then, as the line varies, the point will describe a locus. Thus, 
the centre of a circle passing through two fixed points describes a 
locus while the circle varies. When the co-ordinates of the point 
enter as constants in the equation of the line, the conditions of the 
problem will give a relation between them which will be the equa- 
tion required. For example : 

To find the locus of the vertex of the parabola passing through 
P 1; having a given parameter, and its axis parallel to the axis of X.. 
The equation of the parabola is 

(y-yy = 2p(x-x'), 

in which x! and y' are the co-ordinates of the vertex. By the con- 
ditions of the problem 2p is constant, and 

(y 1 -y'y = 2p(x 1 -x f ). 



* Just as the position of a point, which is determined by two quantities 
x and y, is wholly indeterminate when they are both arbitrary; but when 
they are connected by a single equation the point is conditioned or re- 
stricted, though not determined. 



POINT CONNECTED WITH A VARIABLE LINE. 251 

This is a relation between x\ y' and the constants 2p. x x and y x ; 
hence if we regard x' and y' as variable co-ordinates, it is the equa- 
tion of the locus of P'. Writing x and y in place of x' and y, we 
may put the equation in the form 

O — Mif = — 2p(x — x x ). 

Hence the locus is an equal parabola, extending in the opposite 
direction, and having P x for its principal vertex. 

The number of conditions of the problem is one less than that 
which would determine the parabola. If there were another con- 
dition the vertex would be fixed; if there were one less, it would 
be free to take any position. 

341. If instead of a constant value of 2p, we have another con- 
dition, it will be necessary to determine 2p in terms of x'. y' and 
constants, and substitute its value before changing x' and y' to x 
and y. For instance, if the value of the parameter corresponding 
to the diameter passing through Pj is to be constant, instead of that 
of the principal parameter, we have by Arts. 152 and 147, 

2/ = 2p(l + i?) = 2p + 4X, = 2p +-4fa - x'). 

Hence 2p =. 2p' — 4:(x x — x'). in which 2p' is constant; substitut- 
ing this value in the equation of condition, we have 

(^i -y'Y = %>' (*, - x>) — 4^ - xj. 

Replacing x! and y' by x and y, the locus of the vertex is 

(3/ -yd' + K* - *i) 2 + 2/ (x - x,) = 0, 

an ellipse passing through P r Its equation when referred to Pj as 
origin is 4x 2 -f- y 2 -j- 2p'x — 0, from which it may be shown that 
the centre is the point ( — \p\ 0) and the semi-axes are A = ip', 
B = \p\ the major axis being perpendicular to the axis of X. 

342. When the conditions of the problem give a relation between 
constants which appear in another form of the equation of the vari- 
able line, the equation must be reduced to that form, and the values 
of the constants substituted in the equation of condition. Thus : 

Find the locus of the point whose polar relatively to the ellipse 
Ay + B 2 * = A 2 B 2 is tangent to the ellipse A n y -f B'V =A' 2 B' 2 . 



252 GEOMETRICAL LOCI. 

The equation of the polar of P x is A 2 ^ -j- Wxx x = A 2 B 2 , and 
the condition of tangency for a line in the form y = mx -j- b, is 
(Art. 190) b 2 = A!hn 2 -j- B /2 . Reducing the equation of the polar 
to the form y = mx -f- b, we find values of m and b, which we 
substitute in this equation of condition. The result is 

* = A' 2 ^ + B' 2 , 

^ A^ 2 ^ 

a relation between the co-ordinates of P 1} expressing the condition 
that its polar shall touch the ellipse Mhf + B' V = A' 2 B' 2 . Put- 
ting x and y in place of these co-ordinates, and clearing of fractions, 

A 4 B 4 = A' 2 B 4 x 2 + B' 2 Ay . 

A 2 B 2 

The locus is therefore an ellipse whose semi-axes are — and — ; 

that is, third proportionals to the corresponding semi-axes of the 
given ellipse, and the ellipse of reference. 

Examples. — Find the locus of the point whose polar passes 
through a given point. 

Show that, when the polar of P l with reference to any conic 
touches an ellipse, the locus of P x is a conic. (Assume the conic 
of reference in the general form.) 

Show that the locus of the centre of the conic passing through 
the intersections of two conies is generally of the second degree ; 
but that it is of the first degree when the conies are similar. 

Use of Polar Co-ordinates. 

343. Polar co-ordinates may be used with advantage when the 
point which describes the locus is connected with a fixed point by 
a line of variable length. For, if we make the fixed point the pole, 
and express the length of the connecting line (which will be the 
radius vector) in terms of its inclination, the result will be the polar 
equation. Thus : 

A triangle has a fixed base ; find the locus of its vertex, when the 
value of the vertical angle is given. 

Let AB, the fixed base, be the initial line, and A the pole, then 




USE OF POLAR CO-ORDINATES. 253 

the side AP is the variable radius vector, and the angle at A is 6. 

Denote the constant value of the angle at 

P by a, and the distance AB by a. In 

order to express the value of r in terms of 

and the constants, drop a perpendicular 

from B on AP. It divides r into two parts, 

AR and PR, of which AR = a cos and 

PR == RB cot a = a sin 6 cot a. Therefore 

r = a cos 6 -j- a cot a sin 6. 

Comparing this with the polar equation of Art. 112, we see that 
it represents a circle passing through the pole A (because /— 0), 
and having its centre at the point whose rectangular co-ordinates 
are (?a, \a cot a). 

344. We noticed in Art. 69, that the constants which enter into 
an equation of the form x cos a -f- y sin a =p are the polar co-ordi- 
nates of the foot of the perpendicular from the origin. When the 
equation of a variable straight line is put in this form, we have, in 
place of p, its value in terms of a. Hence, if we write r in place 
of p and 6 in place of a, we shall have the radius vector of the foot 
of the perpendicular expressed in terms of its angular co-ordinate 6; 
that is, the polar equation of the locus of this point* An instance 
of this method of finding a locus is given in Art. 172, in which the 
locus of the foot of the perpendicular from the focus upon a tangent 
to the parabola is found. Applying the same method to the equa- 
tion of Art. 171, we find for the locus of the foot of a perpendicu- 
lar from the vertex 

r cos 6 = — \p sin 2 6. 

To ascertain the degree of this locus, it is necessary to transform it 
to rectangular co-ordinates by the formulae of Arts. 86 and 97. 
Thus, multiplying both members by r 2 and transforming, 

(z 2 +/)* + ^ 2 = 0, 

an equation of the third degree, and the same that would have 
been found, by using the equation of Art. 147 and that of a per- 

* In general, when p is a function of a, let p = F.a, then the equation of 
the variable line is x cos a -f- y sin a = F.a, and the locus of the foot of 
the perpendicular is r = F.0. 
22 



254 GEOMETRICAL LOCI. 

pendicular through the origin, and finding the locus of their inter- 
section by the method of Art. 337. 

345. If we put r equal to the perpendicular from any other point 
instead of the origin, the result will be the locus of the foot of the 
perpendicular from that point. Thus : 

Find the locus of the foot of a perpendicular from (a, 6) on a 
tangent to an ellipse. 

The equation of the tangent, Art. 193, is 

x cos a -J- y sin a = V A 2 cos 2 a -\- B 2 sin 2 a. 
The perpendicular from (a, 5) is 

\/A 2 cos 2 a -f- B 2 sin 2 a — a cos a — & sin «. 
Putting r equal to this quantity, and writing in place of a, 



r = V A 2 cos 2 -f B 2 sin 2 — a cos — b sin 0, 



or V A 2 cos 2 -1- B 2 sin 2 == r -±- a cos ■ + '.6 sin 0. 

Multiplying through by r and transforming, 

l/AV + Byr=x 2 -j- y 2 -f az -f- %* 

Squaring this equation, we shall find it to be, in general, of the 
fourth degree: but it will be divisible by x 2 -j- y 2 , when a 2 = c 2 
and b = ; that is, when the point is one of the foci. In this 
case, but in no other, will it reduce to the second degree. 

Examples. — A straight line passing through a fixed point A on 
the circumference of a circle meets the circle again in B, find the 
locus of a point P moving in the line, so that AB X AP = c 2 . 

Pind the locus when A is not on the circumference. (Take the 
diameter through A as initial line, then the value of AB is one of 
the values of r in Art. 114.) 

Find the locus of the foot of a perpendicular from the origin 
upon the line passing through P'. 

* This is the general equation of the curve, referred to the point (a, b) 
itself as origin. Since it is satisfied by £ = 0, y = 0, it would seem that 
the curve always passed through the point, which is impossible when the 
point is within the ellipse. The reason is that we introduced the root 
r = by multiplying through by r. 



CHAPTER X. 

APPLICATION OF ANALYSIS TO SOLID GEOMETRY. 

346. We observed in Art. 6, that to determine the position of a 
point on a given surface two co-ordinates or determining quantities 
are necessary and sufficient ; but that to determine position generally 
three are necessary on account of the three dimensions of space. 
The investigation of questions involving the position of points not 
all in the same plane is therefore called Geometry of Three Dimen- 
sions, or Solid Geometry, because the discussion of solids requires 
the consideration of the three dimensions, while that of plane figures 
involves only two. 

It is the object of the present Chapter to explain a method by 
which the analytical treatment is adapted to Solid Geometry. The 
systems of co-ordinates used are extensions of those we have already 
applied to Plane Geometry. 

347. Let OX and OY be two co-ordinate axes taken in a fixed 
plane of reference ; the position of any point in this plane is then 
determined by the co-ordinates x and y. Through the origin let a 
line OZ be drawn, not in the plane YOX. If now from any point 
P a line be drawn parallel to OZ and piercing the plane YOX in 
Q, the length of PQ, together with 

the co-ordinates of Q as referred \ 

to the axes OX and OY, deter- ... --V p """' : \. 

mines the position of P. There- \ V \" 

fore, denoting PQ by z, QR by y, \ \ ...Vz 

and OR by a;, we may regard P \ - A--"" x V \ — x 

as determined by the values of \s~.. 1. -V"» 

three co-ordinates, x, y and z, ^^ \ 

which are distances measured in 

the directions of three fixed lines or axes, OX, OY and OZ, called 
respectively, the axis o/X, the axis o/Y and the axis o/Z. 

255 



256 SOLID GEOMETRY. 

If from P a line be drawn parallel to the axis of X, meeting the 
plane YOZ in Q', PQ' and OH will be equal, because they are parts 
of parallel lines intercepted between the parallel planes PQR and 
YOZ. Therefore x represents PQ', the distance of P from the plane 
YOZ, measured in a direction parallel to the axis of X. If a lioe be 
drawn parallel to the axis of Y to meet the plane ZOX in Q", PQ" 
and QK will be equal, and each may be represented by y ; therefore 
the co-ordinates x, y and z may be considered to denote the distances 
of P from three co-ordinates planes, measured from each in a direc- 
tion parallel to the intersection of the other two. These planes are 
called respectively the plane o/'XY, the plane o/YZ and the plane 
o/XZ. The first contains the axis of X and that of Y, the second 
that of Y and that of Z, the third that of X and that of Z. 

348. The plane of the lines PQ' and PQ" is parallel to the plane 
of XY, that of PQ and PQ" to the plane of YZ, and that of PQ 
and PQ' to the plane of XZ. The intersections of these planes 
and the co-ordinates planes, therefore, form the edges of a parallelo-* 
pipedon, or solid whose faces are parallelograms. For every point 
in the first of these planes, the value of z is the same; if this con- 
stant value be denoted by c, then the equation z = c indicates that 
the point P is situated in this plane. It is, therefore, said to be 
the equation of the plane. In like manner, the other planes are rep- 
resented by equations of the form x = a and y = b. The point whose 
co-ordinates are a, b and c, or as we may express it the point (a, Z>, c), 
is therefore the intersection of the three planes x = a, y = b, 
z = c, each of which is parallel to one of the co-ordinate planes. 

In constructing a point with given co-ordinates, it is evident 
that we may lay off the values a, b and c in their proper directions, 
and in any order we please. Thus, we may lay off the value c on 
the axis of Z, and so determine the plane z = c ; then in this 
plane we may construct the point (a, b) using as axes its intersec- 
tions with the planes of XZ and of YZ; that is, the lines OQ" and 
OQ', which are parallel to the original axes of X and Y. 

It is of course necessary to assume a positive direction on the 
axis of Z, as well as on the axis of X and on that of Y. In repre- 
senting position in space it is usual to draw only the positive direc- 
tions of the axes, and to indicate the position of a point, by draw- 
ing the co-ordinates PQ and QK, parallel to the axes of Z and Y. 



CO-ORDINATES OF DIRECTION. 257 



Co-ordinates of Direction. 

349. When the position of a point in a given plane is determined 
by its distance and direction from a fixed point, only one angular 
co-ordinate is necessary, because we have only to compare the 
directions of lines in a single plane. But for points in space two 
co-ordinates of direction are necessary ; one to determine a plane 
containing the line whose direction is to be expressed, and the other 
to fix the direction of the line in that plane. 

Let be the fixed point of reference, and OZ a fixed straight 
line passing through it. It will be convenient to conceive of this 
line as directed upward from 0, or toward the zenith. A plane 
passing through OZ will then be a vertical plane. Suppose such a 
plane to rotate about OZ as an axis, and let P be the point whose 
position is to be determined. It is evident that the vertical plane 
may be made to pass through P, and therefore to contain the line 
OP. 

Let OH be a perpendicular to OZ, drawn in the vertical plane; 
then in the rotation OH will describe a horizontal plane. Assume 
an initial line in the horizontal plane, then the direction of OH in 
this plane may be expressed, as in the system of polar co-ordinates, 
by 0, denoting its inclination to the initial line. The angle 6 may 
therefore be used to determine the position of the vertical plane 
containing OP. Let cp denote the inclination of OP to OH mea- 
sured in the vertical plane ; then the angles d and <p together deter- 
mine the direction of P. 

350. The co-ordinates of direction, 6 and y>, are usually called 
spherical co-ordinates, because it is found convenient, in the treat- 
ment of questions where distance is not considered, to refer direc- 
tion in space to position upon the surface of a sphere. Thus, sup- 
pose the point 0, of the last Article, to be the centre of a sphere 
of any radius we chose ; the line OP, produced if necessary, will 
pierce the sphere in some point, and the point P is said to he pro- 
jected on the surface of the sphere at that point. The position of 
this point is then used to express the direction of OP, or apparent 
position of P as seen from 0. The horizontal plane described by 
OH will cut the sphere in a great circle, which is the primary circle 
of reference. The axis OZ pierces the surface of the sphere in 

22* 



258 SOLID GEOMETRY. 

points which are called the poles of this circle. Now it is evident 
that is equivalent to an arc of the primary circle measured from 
the point in which the sphere is pierced by the initial line, which 
is the origin of the spherical co-ordinates. The vertical plane cuts 
the sphere in a great circle passing through the poles of the pri- 
mary circle, and <p is measured by the arc of this circle included 
between the projection of P and the extremity of the arc 0. 

351. Let the angle <p be measured upward. If the value of is 
fixed, <p — corresponds to a point in the horizontal plane ; if <p 
increase from zero, the line OP moves upward, its inclination to the 
horizontal plane increasing, until <p = 90°, when it becomes perpen- 
dicular to that' plane and coincides with the vertical line OZ. If 
<p increase beyond 90°, the inclination of OP to the horizontal 
plane is diminished, being measured by 180° — <p • but if we add 
180° to the value of the direction of OP will be expressed by a 
value of <p less than 90°. In other words, 180° -f and 180° — <p 
determine the same direction as and <p • hence if we allow all 
values between 0° and 360°, <p may always be taken less than 90°. 
For a point below the horizontal plane, <p is negative; and $>■== — 90° 
makes OP coincide with OZ produced, whatever be the value of 0. 
The limiting values of <p are therefore -f- 90° and — 90°, and* these 
extreme values of <p determine the direction of a line without the 
aid of the co-ordinate 0. 

The length of OP, together with <p and which determine its 
direction, constitutes a system of polar co-ordinates for space. The 
plane in which is measured is called the primitive plane. The 
line OZ perpendicular to the primitive plane is called its axis. 
The initial line may be taken anywhere in the primitive plane ; its 
co-ordinates of direction are = 0° and <p = 0°. Any two straight 
lines which meet at right angles may be made the axis and initial 
line of a system of polar co-ordinates. 

Polar and Rectangular Co-ordinates. 

352. If the primitive plane of polar co-ordinates be taken as the 
plane of XY, and the axis or perpendicular OZ, as the axis of Z, 
then the co-ordinate z will be a perpendicular from P to this plane. 
Let Q be the foot of the perpendicular. This point is called the 
projection of P on the plane, and the straight line OQ, joining Q 



POLAR AND RECTANGULAR CO-ORDINATES. 



259 



with the origin, is called the projection of OP. We may then say 
that <p is the angle between OP and its projection on the primitive 
plane, and that is the angle between this projection and the initial 
line. Denoting the length OQ by r, r and are the polar co-or- 
dinates of Q in the primitive plane. If now rectangular axes be 
assumed in this plane, the axis of X coinciding with the initial 
line, it is evident that each of the axes is perpendicular to the 
plane of the other two. In this case, the axes of X, Y and Z are 
said to form a -rectangular system. This system is universally 
used in the applications of Analysis to Mechanics and Astronomy, 
on account of its connection with the polar system. 

353. In the figure, ROQ is the angle 0, and QOP is the angle <p. 
Denote the distance OP by p ; we 
have now to find the relations 
between the polar co-ordinates 
/>, <p and 0, and the rectangular 
co-ordinates x, y and z. 

By the relations between the 
rectangular and polar co-ordinates 1 
of Q in the plane of XY, we have 

x = r cos and 





z 






Q" 

X 


Q' 







P 

z 




^^..Z^r^r^ 





y = r sin 0. 



In the plane ZOPQ, p and <p may be considered as the polar co-or- 
dinates of P ; while r and z are its rectangular co-ordinates, because 
PQO is a right angle. Therefore 

r = p cos <p and z = p sin <p. 

Hence, eliminating r from the values of x and y, we find 

x = p cos <p cos 0, 
y == p cos <p sin 0, 
z = p sin <p. 

354. If through P planes be passed parallel to the three co-or- 
dinate planes, the intersections will form the edges of a rectangular 
parallelopipedon as represented by the dotted lines in the figure. 
The edges PQ, PQ' and PQ" are the distances of P from the planes of 
XY, YZ and XZ respectively, because they are perpendiculars to 
these planes. Therefore the co-ordinates x, y, z, in the rectangular 



260 SOLID GEOMETRY. 

sybteni, are the distances of P from the co-ordinate planes. The 
diagonal, OQ, which we have denoted by r is the distance of P 
from the axis of Z. By the right triangle QOR, we have r 2 = x 2 -\-y 2 . 
In like manner, if we denote OQ' and OQ" by / and /', r' 2 =y 2 -|- z 2 
and r ff2 = z 2 -j- x 2 . Therefore the distances of P from the axes of 
X, Y and Z respectively are 

Vy 2 + z 2 , lA 2 -f x 2 , Vx 2 + y 2 . 

The right triangle POQ gives p 2 = r 2 -j- z 2 , therefore for the dis- 
tance of P from the origin, we have 

p = Vx 2 -\-y 2 + z 2 . 

Example. — Find the distance of the point (3, 4, 12) from 
each axis and from the origin. 

Method of Projections. 

355. The relations between points and lines in space are most 
conveniently established by the aid of the method of projections, 
which is now to be explained. 

A point is said to be projected on a straight line at the foot of a 
perpendicular from the point to the straight line. The distance 
between the points in which the extremities of a given line are 
projected on a fixed line of indefinite length, is called the projection 
of the given line upon the indefinite line. Thus, CD is the pro- 
jection of AB upon the horizontal line in the 
figure. If AB and the line on which it is pro- 
jected are in the same plane, then the project- 
ing perpendiculars, AC and BD, are also in 
this plane. Draw BE parallel to CD, then 

ABE is the inclination of AB to the indefinite j ; 

line, and EB is equal to its projection. Now 
in the right triangle ABE, 

EB == AB cos ABE ; 

that is, the projection of a line is equal to its length multiplied by 

the cosine of its inclination to the line on which the "projection is made. 

The projection of a line cannot be greater than the line itself, 



METHOD OF PROJECTIONS. 2G1 

because the cosine of an angle cannot exceed unity. The projection 
upon a parallel line is equal to the line itself, and the projection 
upon a perpendicular line is zero. 

356. If EF be drawn perpendicular to AB. BF is the projection 
of EB on AB ; therefore BF = EB cos ABE = AB cos 2 ABE. 
Mow AE is the projection of AB upon AC. and AF is the projec- 
tion of AE upon AB ; hence AE = AB cos BAE, and AF = 
AB cos 2 BAE. Adding the values of BF and AF, we have AB = 
BF + AF = AB (cos 2 ABE -f- cos 2 BAE) ; therefore cos 2 ABE + 
cos 2 BAE == 1. Since the angles ABE and BAE are complements, 
the above is a method of proving the fundamental equation of 
trigonometry, sin 2 -j- eos 2 ==l. If we multiply both members by 
AB 2 , we have (by substituting EB for AB cos ABE. and AE for 
AB cos BAE) EB 2 -f AE 2 = AB 2 , the fundamental relation be- 
tween the sides of a right triangle. This relation may be expressed 
thus : If a line in a given plane be projected upon two perpendicu- 
lar lines of the plane, the sum of the squares of the projections will 
equal the square of the line. 

357. The projection of AB upon any line, whether in the same 
plane with it or not, may be made by passing planes perpendicular 
to this line through A and B. The part of the line intercepted 
between the planes will be the projection. The projections of a 
line upon two parallel lines are equal, each being the distance be- 
tween the same two parallel planes. Since all parallel lines have 
the same direction, a line is considered as having the same inclina- 
tion to a line which does not intersect it as to a parallel line which 
does intersect it. Hence the projection of a line is in all cases 
equal to its length multiplied by the cosine of its inclination. 

The projections of equal and parallel lines upon the same line 
are evidently equal. 

358. Suppose now the projections to be made upon a fixed line, 
and let one direction, measured upon the fixed line, be regarded as 
positive. Let any two points, A and B, be joined by a straight 
line, and also by a broken line ACDB, the. intermediate points C 
and D having any positions whatever. Now, if by the projection 
of AB we mean the distance from the projection of A to that of B, 
considering both the length and direction of this distance, then the 
projection of AB may be positive, or it may be negative. It will 



262 SOLID GEOMETRY. 

be negative, when the direction from A to B makes an obtuse angle 
with the positive direction assumed on the fixed line. The ratio 
of a line to its projection will still be that of unity to the cosine of 
this angle, because the cosine of an obtuse angle is negative. 

If now we consider the signs of the projections, it is easy to see 
that the projection of the straight line AB is equal to the sum of the 
projections of the parts of the broken line ACDB j that is, of 
AC, CD and DB. 

359. A point is said to be projected on a plane at the foot of a 
perpendicular from the point to the plane. If all the points of a 
given straight line be projected upon a plane, the projecting per- 
pendiculars will lie in a single plane. This plane may be called 
the projecting plane of the given line ; it is perpendicular to the 
plane on which the projection is made, and cuts it in a line which 
is called the projection of the given line. Therefore the acute angle 
between a line and its projection is the inclination of the line to 
the plane of projection. The projection of a line of definite length 
upon a plane is therefore equal to its length multiplied by the co- 
sine of its inclination. 

The projections of a given line upon a perpendicular line and 
plane are in reality projections upon two perpendicular lines, be- 
cause a line perpendicular to a plane is perpendicular to every line 
in the plane. Therefore the sum of the squares of these projections 
is equal to the square of the line, by Art. 356. 

Direction Angles' 

360. According to the preceding definitions, the co-ordinates x, 
y and z in the rectangular system are the projections of OP or />, 
the distance of P from the origin, upon the axes of X, Y and Z 
respectively. Far the plane PQR, in the figure, is perpendicular 
to the axis of X, therefore the distance Oil cut off on the axis is 
the projection of OP ; and in like manner the other co-ordinates 
are equal to distances cut off on their respective axes by perpen- 
dicular planes passing through P. 

Let a denote the angle between OP and the positive direction 
of the axis of X, let /? denote that between OP and OY, and y that 
between OP and OZ. Then, regarding p as positive, we shall have 



DIRECTION ANGLES. 



263 



x = p cos a, 



y = p cos /?, z = p cos ^. 



The direction of the line OP passing through the origin is deter- 
mined by the values of these angles; because, if they are known, we 
may assume a value for p and then determine a point on the line, 
which with the origin fixes the line's direction. 

We saw in Art. 349, that two angles are sufficient to determine 
the direction of a line in space. The direction of a given line 
might be defined by means of two angles — namely, its inclination 
to its projection in the plane of XY and the inclination of that pro- 
jection to the axis of X. For a line passing through the origin, as 
OP, these angles would correspond to the polar co-ordinates, <p and 
0, of a point of the line. But more symmetrical results are obtained 
by introducing into our equations a, /? and y, as the direction angles 
of a given line, although they are not three independent quantities, 
but are connected by a relation which we are now to find. 

361. The co-ordinates of the point P, in the figure, OR, RQ and 
QP, form a broken line joining 
the points and P; therefore, 
as explained in Art. 358, the 
sum of their projections upon 
any line is equal to the projec- 
tion of OP on the same line. 
The sum of the projections of 
the co-ordinates upon the line 
OP is therefore equal to OP or p. The projection of x upon OP 
is x cos a, because a is its inclination to OP. Hence, the value of 
this projection is p cos 2 a. In like manner, the projection of y is 
p cos 2 /?, and that of z is p cos 2 y. Hence p cos 2 a -\-p cos 2 /? -|- 
p cos 2 y = p, or 

cos 2 a -|- cos 2 /9 -f~ cos2 Y = 1- 

This is the fundamental relation between the three direction 
angles a, /5 and y* If we multiply each member by p 2 : and put 




* These direction angles may be represented by arcs of great circles upon 
the surface of a sphere, in the following manner : The origin being the 
centre of the sphere, let X, Y and Z be the points in which the positive 
axes pierce the surface; then XYZ is a tri-rectangnlar and tri-qundrantal 
triangle ; that is, its angles are all right angles, and its sides quadrants. 



264 SOLID GEOMETRY. 

x, y and z in place of their values, we obtain the equation 

^ 2 + y 2 + 2 2 = i° 2 ; 

that is, the sum of the squares of the projections of a line upon three 
perpendicular lines equals the square of the line. 

We have thus found by the method of projections the relation 
between the distance from the origin and the rectangular co-ordi- 
nates of a point, that was found in Art. 354. 

362. The quantities cos a, cos ft and cos y are called direction 
cosines. They are the factors by which we multiply the length of 
a line to produce its projections upon the axes, and they may have 
any values, of which the sum of the squares is unity. Thus, -f-, ^ 
and ^ are the direction cosines of a certain line. It is not neces- 
sary to attribute them to a line passing through the origin, if, as 
explained in Art. 357, parallel lines are considered as having the 
same inclinations to the axes. If a line makes an obtuse angle with 
one of the axes, the corresponding direction cosine is negative. It 
is evidently only necessary to consider values of the direction 
angles less than 180°. 

The arcs PX, PY and PZ will then measure the direction angles, a, (3 and 
y, of the radius OP. 

Produce ZP to meet XY in H; then because Z is the 
pole of XY, ZP is the complement of PH, and cos 2 ZP -f- 
cos 2 PH = 1. For a similar reason, cos 2 XH-j-cos 2 HY=1, 
therefore cos 2 ZP + cos 2 PH (cos 2 XH + cos 2 HY) == 1. 
But the arc PH is perpendicular to XY, hence, by a 
formula of spherical right triangles, cos PH cos XH =cos PX 
and cos PH cos HY — cos PY. Therefore by substitution, cos 2 ZP + 
cos 2 PX -f cos 2 PY= 1, which is the above fundamental formula. % 

The arcs PH and XH are the spherical co-ordinates and 0, of the point 
P, or the angular co-ordinates of the direction of OP. (See Art. 350.) 
Comparing the values of x in Arts. 353 and 360, we have cos <p cos 6 = cos a, 
which is a proof of the formula of spherical trigonometry above referred to. 
The axes in the figures are so taken, that increase in the angle 6 (which is 
measured from the axis of X toward that of Y) corresponds to positive rota- 
tion about the axis of Z. If the axis of Z is directed upward, this rotation 
takes place in a horizontal plane, and by common consent it is assumed to 
be positive, when in the order of the cardinal points of the compass N.E.S.W., 
or in the direction of the hands of a watch. The arrows indicate the direc- 
tion of positive rotation about each of the axes. 




DIRECTION ANGLES. 265 

If one of the direction angles is 0°, the line is parallel to the 
corresponding axis, and each of the other direction angles is 90°. 
Thus, if a = 0°, it is parallel to the axis of X, therefore ft == 90° 
and y = 90°; the direction cosines in that case are cos a = l, 
cos ft = 0, cos y = 0. The values a = 180°, /9 = 90°, r = 90°, 
or cos a = — 1, cos /9 = 0, cos y = 0, correspond to the negative 
direction of the axis of X. If one of the direction angles is 90°, 
the line is in a plane perpendicular to the corresponding axis. Thus, 
if y = 90°, it is in a plane parallel to that of XY. 

If two of the direction cosines of a line and the sign of the third 
are known, the direction is determined. Thus, given cos a = -| 
and cos /? = 1 ; by the fundamental formula we obtain cos 2 y = -J, 
hence cos y = ± § ■ Therefore the direction is not determined un- 
less we know the sign of cos y. If we change the signs of all three 
of the direction cosines at once, we obtain those which belong to 
the opposite direction. Thus, |-, — |- and -f- indicate one direction 
in a certain line, and — |-, -J, — |- indicate the opposite direction 
in the same, or in a parallel line. 

363. To find the angle between two lines of which the direction 
cosines are given. 

Let a, /5 and y denote the direction angles of one of the lines, 
and a', ft and /, those of the other. Let d denote the mutual in- 
clination of the lines, which we will at first suppose to pass through 
the origin. Let p denote any length OP measured off from the 
origin on the first line, then the co-ordinates of P are by Art. 360, 
x = p cos a,i/ = p cos /?, z = p cos y. The projection of p upon the 
second line is the sum of the projections of x, y and z (Art. 358). 
Since d is the angle between the lines, the projection of p is p cos 8. 
Since a' is the inclination of the axis of X to the second line, the 
projection of x is x cos a'. In like manner, -the projection ofy is 
y cos ft, and that of z is z cos /. Therefore 

p cos d = x cos a' -j- y cos ft -j- z cos y' . 

Substituting the values of cc, y and z, and dividing by />, we obtain 

cos S = cos a cos a' -j- cos /? cos ft -j- cos y COS /. 

Since parallel lines have the same direction angles, this result applies 
to lines which do not pass through the origin, as well as to those 

23 M 



266 SOLID GEOMETRY. 

which do. The value of d may be found from its cosine by means 
of the trigonometric tables. 

Examples. — Find the cosine of the angle between the lines 
whose direction cosines are respectively J, 1, — |-, and -J-, — -§-, — -|. 
(The direction cosines are always given in the order afty.) 

Find two values of the third direction cosine, the values of the 
first two being -J and f ; and find the cosine of the angle between 
the resulting directions. 

Show that the formula gives the supplemental angle when one 
of the directions is reversed ; also that, if the directions are the same, 
the result is <5 = 0°. 

What is the value of cos <5, when both the given lines are in the 
plane of XY ? (For a line in this plane cos y = 0, therefore by 
the fundamental formula cos 2 a -\- cos 2 ft = 1 or cos ft = ± sin a.*) 

What is the value of cos <5, when one of the lines is in the plane 
of XY and the other in that of XZ ?f 

364. If d = 90°, cos d = ; therefore the condition that two 
lines shall be perpendicular is 



cos a cos a' -\- cos /? cos ft -f cos y COS / = 0. 

Let a, b and c represent the direction cosines of a known line, and 
q, r, s, those of an unknown line ; then, by the fundamental formula 
of Art. 361, we have the equation q 2 -\- r' 1 -j- s 2 = 1, and if the 
line is to be perpendicular to the given line, aq -j- br -f- cs — 0. 
We have, therefore, but two equations between three unknown 
quantities. Since three equations will determine the values of q, r 
and s, a line may be found perpendicular to two given lines. For 
example, let the direction cosines of one of the given lines be ^, ^, -§-, 
and those of the other if, — i- tV ^he three equations to be 
solved are 

q 2 + r 2 -{- s 2 =l, 

2q+ r -f 2s = 0, 

and 14^ — 5r -j- 2s = 0. 

* The result is the formula of plane trigonometry for the difference or 
for the sum of two angles, according as a and a f are measured in the 
same or in opposite directions. 

f The result is a formula of spherical right triangles, since the planes 
are perpendicular. 



TRANSFORMATION OF CO-ORDINATES. 267 

From the two equations of the first degree, we obtain by elimina- 
tion, r = 2q and s '== — 2q. Substituting these values in the 
quadratic equation, we find q = ± ^ ', therefore r = ± ^ and 
s =.qz j. The problem has two solutions, which, however, only 
express the two opposite directions in the same straight line. 

Examples. — Show that j^-, — tt, tt au( ^ TT' ~it- — tt are tr ^ e 
direction cosines of two perpendicular lines j and find those of a 
line perpendicular to both. 

Given — f> T' f an d y> y> — y? show that the lines are perpen- 
dicular, and find a line perpendicular to both. 

Transformation of Co-ordinates. 

365. In passing from one system of co-ordinate axes to another, 
we may consider separately the case in which the origin is changed 
and that in which the directions of the axes are changed. 

Let P', the point whose co-ordinates are cc', y, zf, be the origin 
of a new system of co-ordinates, in which the directions of the axes 
are unchanged. The new plane of XY is parallel to the old. and 
will divide the co-ordinate z of any point into two parts, one of 
which is z', and the other is the corresponding new co-ordinate of 
the point. Denoting this by Z we have z = z' -\- Z. Treating 
the other co-ordinates in the same manner, we obtain, as extensions 
of the formulae of Art. 85, 

x = X + at, y=Y-j-y, * = ?.+ *'• 

We may also express the new co-ordinates in terms of the old, thus : 
X = x-x', Y=y-ij\ Z=.z-z'. 

Of course the values of the constants x\ y' and z! may be negative, 
as well as those of the variables, these expressions denoting algebraic 
sums and differences. 

Examples. — What does the equation x 2 -f if -f- z 2 = 81 be- 
come, when the origin is placed at the point (4, 7, — 4) ? 

What are the new co-ordinates of (1, — 3, — 2), when referred 
to the origin (— 2, 2, — 1) ? 

366. Now suppose the old axes to be rectangular and the origin 
to be unchanged. Let a, b and c be the direction cosines of the 



268 SOLID GEOMETRY. 

new axis of X ; a f y b', c', those of the new axis of Y; and a", b", c", 
those of the new axis of Z. Denote the new co-ordinates of the 
point P by X, Y and Z, and the old co-ordinates by x, y and z. 

Now, whether the new axes be rectangular or oblique, the co- 
ordinates X, Y and Z form a broken line connecting P with the 
origin (see Fig. Art. 347). Therefore the projection of OP upon 
any line is equal to the sum of the projections of X, Y and Z. 
Since the old axes are rectangular, x is the projection of OP upon 
the old axis of X. Since a is the cosine of the inclination of the 
new axis of X to the old, the projection of the co-ordinate X is orX. 
In like manner, a' being the cosine of the inclination of the new 
axis of Y to the old axis of X, the projection of Y is a'Y. The 
projection of Z is, for the same reason, a"Z j hence x = aX -f~ 
a'Y -\- a"Z. Proceeding in the same manner, we obtain expressions 
for y and z, the projections of OP upon the old axes of Y and Z 
respectively ; the projecting ratios being in the first case b, V and 
&", in the second, c, c' and c". Hence the formulae, 

x = aX + a'Y + a"Z, 
y = bX + b'Y + 6"Z, 
z = cX-\- c'Y -f c"Z. 

The formulae contain nine constants, but they are not all inde- 
pendent, for, since a, b and c are direction cosines, a 2 -\- b 2 -j- c 2 = 1, 
and, for the same reason, a' 2 -f b' 2 -j- c n = 1 and a" 2 + b" 2 + c" 2 = 1. 
Hence the nine constants are connected by three equations, and 
only six of them can be regarded as independent. 

367. If the new axes of X and Y are perpendicular, we shall 
have, by Art. 364, the equation ad -j- W -|- cc' = between their 
direction cosines. Similar equations will express that the new axis 
of Z is perpendicular to that of X, and to that of Y. Therefore, if 
the new axes, as well as the old, are rectangular, the nine constants 
will be connected by the following six relations : 

aa'+bb' + cc' = 0, 
aa "J r hb"^ r cc"=0, 



a 2 ~\- 


b' 1 


+ c 2 = 


1, 


a' 2 + 


b n 


! + c' 2 = 


1, 


a" 2 + 


ynj rQ ,n = 


1, 



Only three of the nine constants can in this case be regarded as 



TRANSFORMATION OF CO-ORDINATES. 269 

independent.* If we should assume values for a, b and «', for in- 
stance, the other six might be determined; but there would be 
several solutions, because three of the equations are of the second 
degree. 

368. Since a, a' and a" are the cosines of the inclinations of the 
three new axes to the old axis of X, they are its direction cosines 
as referred to the new axes. So also b, b' and b" are the direction 
cosines of the old axis of Y, and c, c\ c" those of the old axis of Z. 
Hence, the new rectangular co ordinates expressed in terms of the 
old, are 

~X. = ax-\-by-{-cz. 

Y = a'x-\ r b'y + c'z, 

Z = a"x + b"y + c"z. 

We may also prove, in the same way as in the last Article, the six 

relations, 

a 2 _j_ a n + a m =lj ahJr a > h > _|_ a ny, = ^ 

tf _j_ h n _j_ yn = ^ ac _j_ aV _|_ ^]jt = 0? 

C 2 _|_ c* _j_ c >>* = l j be -f- b'c' + b"c" = 0. 

These new relations between the nine constants are of course conse- 
quences of the six relations first found. 

Examples. — Show that, if the direction cosines of the new axis 
of X avp 2 1 2 ofY l !-i 2 and of Z 2 2 H 

01 JS. die 3. -3, -3, 01 X, -3, -J-g, 7-5, dUQ OI Zi, -3, y-g-, y^-, 

the new axes will be rectangular. Find in this system the new 
co-ordinates of the point (1,2, — 1). 

What does the equation x 2 -(- y 2 -j- z 2 = R, 2 become, when trans- 
formed to a new system of rectangular co-ordinates ? 

Verify the value of X in Art. 368, by means of the formulae of 
Art. 366 and the equations of Art. 367. 

* Formulae for transformation from one system of rectangular co-ordinates 
to another are frequently given, in which the constants employed are the 
functions of three independent angles determining the position of the new 
planes and axes. But it is impossible so to seleet these angles as to avoid 
complicated and unsymmetrical results. In a practical case the several 
direction cosines are readily determined, and in the general discussion of 
equations the greatest advantage is gained by the use of the formula? of Art. 
366. and the twelve symmetrical relations between the constants in Arts. 
367 and 368. 
23* 



270 SOLID GEOMETRY. 

369. If two systems of oblique axes have the same origin, the 
rectangular co-ordinates of a point may be expressed in terms of its 
co-ordinates in each of the oblique systems, by means of the formulae 
of Art. 366. The equality of the values of each of the rectangular 
co-ordinates gives an equation of the first degree between the co- 
ordinates in the two oblique systems. Solving the three equations 
thus found, we might express the co-ordinates of one system in terms 
of those of the other system, and the expressions found would be 
of the first degree. Therefore the general formulae of transforma- 
tion for change in the direction of the axes would be of the same 
form as the equations of Art. 366, but the coefficients a, b, c, etc., 
would have different significations. Now these formulae, and also 
those of Art. 365, being of the first degree, it is evident that trans- 
formation of co-ordinates cannot raise the degree of an equation be- 
tween x, y and z. Neither can it lower the degree, because the 
reverse transformation, which must reproduce the original equation, 
cannot raise the degree. 

The formulae for transformation from rectangular to polar co-or- 
dinates have been already found in Art 353.* To express the 
polar co-ordinates in terms of the rectangular, we have, from the 
values of x, y, z and r (since r = \/x 2 *{- y' 2 ), 

y z 

tan 6 = ^, tan (p 



Vx 1 + y 2 



and p = yx 2 -\- if -j- z 2 . 



* In passing from one system of polar co-ordinates to another, the polar 
values of the old and new co-ordinates may be substituted in the formulae 
for change of origin, or in those for change of direction of the axes. 
In the latter case, the value of p is unchanged, and the resulting equations 
become (by dividing through by p) relations between two systems of spheri- 
cal co-ordinates. If the primitive planes coincide, the value of <p is un- 
changed, and 6 is measured from a new initial line, therefore the difference 
of the two values of 6 (or their sum, if measured in opposite directions) will 
be known. If the planes of XZ coincide, the value of y is unchanged, and 
the relations between the old and new values of a: and z may be found by 
the formulae of Art. 88 for rectangular transformation in a plane. A simi- 
lar method may be used when the planes of YZ coincide. These planes 
may always be made to coincide, by taking the intersection of the primitive 
planes as the initial line or axis of X. 



equations between co-ordinates. 271 

Equations between Co-ordinates. 

370. The position of a point in space is determined by the values 
of three co-ordinates; and since the values of three unknown quan- 
tities may be determined by three equations, the position of a point 
is in general determined by three equations between its co-ordinates 
x, y and z. Let us now consider the meaning of a single equation 
connecting these quantities. 

It is evident that x, y and z, in a given equation, may take a 
variety of values; those of x andy, for instance, may be assigned 
at pleasure, and a corresponding value of z may be derived from 
the equation. Thus, given the equation 2x — 3y -f- z = 5 ; if we 
assume x = 2 and y = 1, the equation gives z = 4 ; if we assume 
x = 1 and y = 0, it gives z = 3. We may therefore regard z as 
a function of two independent variables, x and y, because it depends 
upon them both for its value. When the value of the function is 
directly expressed in terms of the independent variables, it is called 
an explicit function. Thus, from the above equation we derive 
z = 5 — 2x-\- 3y, in which z is made an explicit function of x 
and y. 

By substituting any assumed values for x and y, a corresponding 
value of z may be obtained, and thus we may determine and con- 
struct any number of points whose co-ordinates satisfy the given 
equation. 

371. To obtain an idea of the situation of the various points 
which satisfy a given equation, we must consider first those which 
have a fixed value for one co-ordinate. We have seen in Art. 348 
that the point (a, 6, c) may be constructed by constructing the 
point (a, h) in the plane z = c, which is parallel to the plane of 
XY. Now if we give to z, in the equation, a certain value, the re- 
sult is an equation between the co-ordinates x and y of all the 
points situated in a certain plane which satisfy the original equa- 
tion. Thus, given the equation x 2 -f- y 2 -f- z 2 = 25 ; if we make 
2 = 3, we have x 2 -f- y 2 = 16. Supposing the axes rectangular, 
this is the equation of a circle whose radius is 4. Therefore a 
circle having this radius, constructed in the plane z = 3, will con- 
tain all the points of that plane which satisfy the given equation. 

In general, if z = c, we have the equation x 2 -f- y 2 = 25 — c 2 , 



272 SOLID GEOMETKY. 

representing a circle in the plane z = <?, whose centre is in the axis 

of Z. The radius of the circle is l/25 — c 2 . If c = 0, the plane 

coincides with the plane of XY, and 

the radius of the circle is 5 ) and if we 

suppose c to increase from this value, 

the plane moves upward, continuing (" / 

parallel to the plane of XY. The "r 

radius of the circle decreases gradually ..-•'" I 

as the plane moves, and its circum- '"•• \/ _ _^-^^ 

ference will describe a surface. When // 
c = 5, the radius of the circle becomes y 

zero ; if c increase beyond that value, the circle becomes imaginary. 
If c be made negative, the radius of the circle will decrease in the 
same manner as we pass from c == to c = — 5, and beyond this 
limit it is again imaginary. It is plain that the surface thus de- 
scribed will contain all the points which satisfy the equation 

X * _|_ y 2 J^_ z 2 -__ 25- 

372. It may be shown in like manner of any other equation con- 
taining x, y and z, that all the points which satisfy it are situated 
upon a certain surface. The curves of which the equations are 
found by giving fixed values to z, are the intersections of this sur- 
face by planes parallel to that of XY : they are called sections of 
the surface. The section by the plane of XY itself is found by 
making z = 0, and is called the trace of the surface upon that 
plane. 

If we give a fixed value to y in the equation, we obtain an equa- 
tion between x and z, which represents the section of the surface 
by a plane parallel to that of XZ ; and if we make y = 0, we ob- 
tain the section by the plane of XZ, or the trace of the surface upon 
that plane. Similar remarks apply to the results of giving fixed 
values to x. In the example, x 2 -j- y 2 -f- z 2 = 25, each of the 
traces is a circle whose centre is at the origin and whose radius is 
five units in length. The centre of the varying circle, which we 
regarded in the last Article as describing the surface, is always in 
the axis of Z ; and we may consider the variation in its radius to 
be regulated by the trace in the plane of XZ, of which a quadrant 
is drawn in the figure. For this reason, this trace is sometimes 
called the director, while the varying curve is the generator of 



EQUATIONS OF THE PLANE. 273 

the surface : in the example, the surface generated is that of a 
sphere. 

373. It was shown in Art. 369 that the degree of an equation 
between x, y and z ; that is, of the equation of a surface, cannot be 
changed by transformation of co-ordinates. Surfaces are therefore 
classified according to the degrees of their equations. Thus, 
2x — y -\- 3z — 6 represents a surface of the first degree, and 
x 2 — 2xy = 4:z — y, one of the second degree. The section of a 
surface by any plane may be found, by transforming to a new sys- 
tem of co-ordinates in which the plane of XY is parallel to the given 
plane, and then giving the proper value to z in the new equation. 
Now it is evident that the degree of the section cannot exceed that 
of the surface. Therefore every plane section of a surface of the 
first degree is a straight line, and every plane section of a surface 
of the second degree is either a conic or a straight line. 

Equations of the Plane. 

374. The general equation of the first degree is 

Ax-fBy-f + D = 0; 

and, because it is shown in the last Article that every plane inter- 
sects a surface of the first degree in a straight line, this is the 
general equation of the plane. 

The traces of the plane represented by this equation upon 
the co-ordinate planes are the straight lines, Ax -\-~By-\-D = O i 
Ax -f Cz -f D = and By -f Cz -f D = 0. The intercepts of 
the plane upon the axes, which are also the intercepts of these 
lines, are 

D D D 

*o-~ A , ^o---, *°--c' 

If D == 0, all the intercepts become zero, and the plane passes 
through the origin ; in fact, whatever the degree of an equation, if 
there is no absolute term it is satisfied by making x = 0, y = 
and 2 = 0, and therefore the surface it represents passes through 
the origin. 

If one of the coefficients A, B or C is zero, so that one of the 
variables is wanting in the equation, the corresponding intercept 

M* 



274 SOLID GEOMETRY. 

becomes infinite. Thus, if C = 0, z = oo j in other words, the 
plane does not cut the axis of Z. Therefore the equation Ax -\- 
By -f- D = 0, regarded as the equation of a surface, represents a 
plane parallel to the axis of Z. This is also the equation of the 
trace upon the plane of XY, but since z does not enter the equa- 
tion, it may have any value we please. The trace on the plane of 
XZ is, in this case, Ax -J- I) = 0, which represent a line in that 
plane parallel to the axis of Z : the trace on the plane of YZ is 
By -|- D = 0, which represents a line in that plane also parallel to 
the axis of Z. 

In like manner, an equation which wants the term By represents 
a plane parallel to the axis of Y, and one without the term Ax, a 
plane parallel to the axis of X. 

375. The equation of the plane is readily expressed in terms of 
its intercepts. Let the given values of the intercepts be a, b and c ; 

then a = whence A = . Finding in the same manner 

A o 

values for B and C, and substituting in the general equation, we 

obtain 

a^ b^ c~ ' 

which is the required equation. 

The general equation contains four constants, whose ratios deter- 
mine the values of the intercepts, and therefore fix the position of 
the plane. These ratios constitute three arbitrary or independent 
constants, whose values may be so determined as to make the plane 
fulfil three conditions. Thus, a plane may be made to pass through 
three given points; as (2, — 1, 5), (3, 2, 1) and ( — 1, 2, — 1). 
For assume the equation in the form x -\- by -\- cz -\- d = Q ) then 
we have three equations of condition, from which by elimination 
we find b = — 3, c = — 2, d= 5, and the required equation is 
x — 3y — 2z -f 5 = 0. 

Examples. — Find the equation of the plane passing through 
(1, 0, — 2), (3, 2, — 1) and (5, — 1,2); and give the values of 
its intercepts. 

Give the equation of the plane whose intercepts are 3, — 1, 
and 6. 

376. When the axes are rectangular, a convenient form of the 



EQUATIONS OF THE PLANE. 275 

equation of the plane is that in which the constants are the length 
of the perpendicular from the origin upon the plane and the direc- 
tion cosines of this perpendicular. Let P be any point of the 
plane, and the origin; and join OP. Letp denote the length of 
a perpendicular to the plane from the origin, and a, [i and y its 
direction angles. The projection of OP upon the perpendicular is 
p, and this is equal to the sum of the projections of x. y and z 
upon the same line. But the projection of x upon this line is 
x cos a, and those of y and z are y cos ,2 and z cos y. Therefore, 

x cos a -\- y cos /? -j- z cos y = p. 

To reduce a given equation to this form, we must divide both 
members by the square root of the sum of the squares of the co- 
efficients of x, y and z, in order to make the coefficients fulfil the 
fundamental condition of direction cosines in Art. 361. For exam- 
ple, given 4x — 7y — 4z = 12, we must divide by 9 : the result is 
4-.r — ly — %z = |- ; therefore, the length of the perpendicular from 
the origin is -J, and its direction cosines are |-, — -J and — |-. If 
the absolute term in the second member is negative, the coefficients 
will be the negatives of the direction cosines of the perpendicular, 
which, as shown in Art. 362. belong to the opposite direction, or 
that of the perpendicular produced. 

Examples. — Reduce to the above form 3x — 5y -f 4z -j- 10 = 0, 
and show that the perpendicular makes an angle of 45° with the 
axis of Y. 

Find the perpendicular from the origin on Qx — 2y — 9z -j- 2 = 0. 

377. The general equation Ax -j- By -J- Qz -f D = is reduced 
to the form x cos a -f- y cos ft -\- z cos y = p by dividing by 
V A'- -\- B 2 -j- C 2 . If the equation contains but two of the variables, 
one of the direction cosines is zero. Thus, if C = 0. cos y = or 
y == 90°, and the perpendicular lies in the plane of XY ; the equa- 
tion in this case takes the form x cos a -f- y sin a =p : which repre- 
sents a plane perpendicular to the plane of XY, or parallel to the 
axis of Z. If the equation contains but one of the variables, two 
of the direction cosines vanish. Thus, if A = and B = 0, 
a = 90° and ;3 = 90°, and the perpendicular coincides with the 
axis of Z ; the equation in this case takes the form zb z =p, which 
represents a plane parallel to the plane of XY. 



276 SOLID GEOMETRY. 

If from a point on the intersection of two planes perpendiculars 
to the planes are drawn, and a plane is passed through them, it 
will cut the given planes in lines perpendicular to the line of inter- 
section. The angle between these lines measures the inclination 
of the planes, and it is evidently the same as that between the per- 
pendiculars to the planes. Therefore the inclination of two planes 
is the same as that of the perpendiculars to the planes. Therefore, 
when the direction cosines of the perpendiculars to two planes are 
found, the angle between the planes may be found by the formula 
of Art. 363. 

378. If two planes are parallel, their traces upon each of the 
co-ordinates planes are parallel, and therefore the coefficients of x, y 
and z in their equations must be proportional. Thus, x -\- 2y — 
2z -J- 1 = and 3x -f- % — 6z = 5 represent parallel planes. 

When the axes are rectangular, the coefficients A, B and C are 
proportional to the direction cosines of the perpendicular; therefore 
the condition that two planes shall be perpendicular (from the 
formula of Art. 364) is 

AA' + BB'-f CC' = 0. 

Thus, the planes x-\- 2y — 2z-{- 1 =0 and 2x -f- hy -\- 6z = 5 are 
perpendicular. 

We saw in Art. 375 that a plane may be made to fulfil three 
conditions. Now, to be parallel to a given plane, or to be perpen- 
dicular to a given line, is equivalent to two conditions, because it 
determines the ratios of the three coefficients A, B and C. But to 
be perpendicular to a given plane is equivalent to a single condition 
imposed upon the coefficients, and two such conditions will deter- 
mine the ratios. Thus, if a plane is to be perpendicular to the 
planes x -f- 2y — 2z -(-1 = and 3x — y -\- 2z = 6, the con- 
ditions give A -f 2B — 2C = and 3A — B -f 2C = 0. By 
elimination, we find B = — 4A and C = — -J A ; therefore the 
coefficients are as the numbers 2, — 8 and — 7. Since the value 
of one of the coefficients may be assumed at pleasure, the required 
equation is 2x — 8y — 7z -j- I) = 0, in which the absolute term 
may be so determined as to make the plane pass through a given 
point. 

379. The general equation of the plane passing through a given 



EQUATIONS OF THE PLANE. 277 

point, P', (found by eliminating D from the general equation by an 
equation of condition) is 

A (x-x f ) + B (y-y f ) + C (z — J)=f. 

This equation is evidently satisfied by the point P\ It might also 
have been found by substituting, in the equation of a plane passing 
through the origin, the values of the co-ordinates of P referred to 
P' as a new origin. For, by Art. 365, x — x\y — y' and z — z' 
are the values of the new co-ordinates; and AX -j- BY -f- CZ = 
is the equation of a plane passing through the new origin. 
In the same manner it may be shown that 

(x — x') cos a -j- (y — y'~) cos /3 -j- (z — z') cos y = 

is the equation of a plane passing through P' and perpendicular to 
the line whose direction angles are a, /? and y. 

380. If Ax + B^-f C2 + D = and A'z 4- B'y + C's + D* = 
are the equations of two given planes, then 

Ax -f By -f Qz -j- D -f k (A!x + B'y -f Q'z -f D') = 

is the equation of a plane passing through the straight line in 
which these planes intersect ; for it is evidently satisfied by all the 
points which satisfy both the given equations, and being of the 
first degree it represents a plane. The value of k is arbitrary, and 
may be so determined as to make the plane fulfil another condition ; 
for instance, that of passing through a given point. It may also 
be so determined as to eliminate one of the variables from the equa- 
tion, thus making the plane parallel to the corresponding axis, as 
shown in Art. 374. 

Examples. — Grive the general equation of the plane passing 
through the intersection of 2x -f- y — 3z -j- 1 = with x — 2y -f- 
z -j- 3 = 0, and determine the plane so as to pass through the point 
(3,2-1). 

Determine the planes passing through the same intersection, and 
parallel respectively to the axes of X, Y and Z. 

Determine the plane passing through the same line, and perpen- 
dicular to the plane Sx -\- 2y -j- 22 = 0. (Substitute the coeffi- 
cients of this equation, and those of the general equation contain- 
ing k, in the condition of Art. 378.) 
24 



278 SOLID GEOMETRY. 



Equations of the Straight Line. 

381. Since a single equation between x, y and z restricts a point 
to a certain surface, two such equations taken together restrict a 
point to the line common to the surfaces ; that is, to their line of 
intersection. Two equations of the first degree, therefore, restrict 
a point to the intersection of two planes ; that is, to a certain straight 
line. 

The position of a straight line is in reality determined by the 
equations of any two planes passing through it. Therefore, if 
Ax + By -f Qz -f D == and A'x + B'y -f C'z + D' = are the 
equations of two planes passing through a given line, any two of 
the series of equations which result from giving different values to 
k in the equation of Art. 380 will determine the same line. The 
equations of simplest form are those which result from eliminating 
one of the variables. Thus, given the equations, 2x -j- y — 3z -j- 
1 = and x — 2y -J- z -f- 3 = ; the simplest equations of planes 
passing through the line determined by them, are x — y -J- 2 = 0, 
x — z -\-l = and y — z — 1 = 0, found by eliminating respect- 
ively 2, y and x. Thus, there are three equations, of equal sim- 
plicity, any two of which would serve to determine the line in 
question and may therefore be considered as its equations. 

382. Generally, two of the variables in the equations of a line 
may be expressed as functions of the third. Thus, in the example, 
the first two of the results of elimination may be written in the 
forms, y r=x-\- 2, z = x-\- 1. (The equation between y and z is 
not independent of these, but may be derived directly from them.) 
Therefore x may be considered as an independent variable, for 
which a value being assumed at pleasure, the corresponding values 
of y and z are determined by the equations. If, for instance, we 
assume x = 1, the corresponding value of y is found to be 3, and 
that of z to be 2, therefore the point (1, 3, 2) is a point on the line, 
as may be verified by showing that it satisfies each of the original 
equations. 

If, however, the line is parallel to one of the co-ordinate planes, 
one- of the co-ordinates is constant; the equations of the line must 
then express the value of this co-ordinate, and a relation between 
the values of the other two. This is the case with the line in 



EQUATIONS OF THE STRAIGHT LINE. 279 



which x — y — 2z -f 5 = and x -\- 2y -\- 4z — 4 = intersect ; 
for if we eliminate z we shall at the same time eliminate y, the result 
being 3x -}- 6 = or x = — 2. The line is therefore situated in 
the plane x = — 2, which is parallel to the plane of YZ. Elimi- 
nating x between the equations, we obtain y -\- 2z — 3 = 0; and 
the equations of the line are x = — 2 and y -\- 2z — 3 = 0. 

Since a straight line is always given by means of the equations 
of two planes passing through it, the formula of Art. 380 may be 
used to find the equation of a plane passing through a given straight 
line and fulfilling another condition. 

Examples. — Determine the plane passing through the line 
y = 2x — 3, 2 = 5 — x and through the point (1, — 1,3). 

Show that, if a line is parallel to the plane of YZ, the traces of 
all planes passing through it on the plane of YZ will be parallel. 

383. To find the equations of the line passing through two given 
points. Let P' and P" be the given points, and let P be any 
point of the line passing through them. Let planes parallel to the 
plane of YZ be passed through P, P' and P". The parts of the 
axis of X cut off by these planes are the co-ordinates x, x' and x" ; 
therefore the parts intercepted between the first and second of 
these planes is x — x\ and that between the third and second is 
x" — x'. The corresponding segments of the line are PP' and P"P'. 
Now the segments of two lines included between parallel planes 
are proportional; therefore x — x' : x" — x' : : PP' : P"P', or 

X X ' PP' y y> PP' 

In like manner we may prove 



x" —x' P"F " x y— y P"F 

z — z' PP' 

and = , Hence 

z"— z f F'P' 

x — x y — y' z — z' 



X " — X > y" _y z" — Z f 

This is equivalent to two independent equations ; the equality of 
the first and second members being the equation between x and y, 
and that of the first and third, the equation between x and z. P' 
satisfies these equations because it reduces each member to zero, 
and P", because it makes each member unity. 

Examples. — Find the equations of the straight line passing 



280 SOLID GEOMETRY. 

through (5, 3, — 1) and ( — 2, 1, 4), and express y and z as func- 
tions of x. 

Grive the general equation of the plane passing through these 
points, and determine k so that the plane shall also pass through 
(1,2,1). 

384. If we put L, M and N in place of x" — x', y" —y f and 
z" — z', we have 

x — x' y — of z — z f 

L ~~ M = N ' 

in which L, M and N may have any values whatever. These are 
therefore the general equations of the straight line passing through 
the given point P'. The direction of the line depends upon the 
ratios of L, M and N , and not upon their absolute values. 

If the point P" coincide with P', the value of each of these quan- 
tities is zero, and the direction of the line is indeterminate. If P" 
and P' have one common co-ordinate, the line is parallel to one of 
the co-ordinate planes. Thus, if x" = x' or L = 0, the numera- 
tor of the first member must be zero, otherwise the fraction takes 
the infinite form. Therefore in this case, x — x f = 0, and the 
equations of the line are 

, . y — y' z — z 
x = x' and - — — = , 

M N ' 

of which the first expresses that the line is in a plane parallel to 
the plane of YZ ; therefore it cannot meet that plane; in other 
Words, it is parallel to it. 

If P' and P" have two common co-ordinates, the line is parallel 
to one of the axes. Thus, if x" = x' and y" =y f , that is, if L = 
and M = 0, we must have 

x = x f and y = y' . 

The equations of the line, therefore, express that it is the inter- 
section of two planes parallel respectively to the plane of YZ and 
to that of XZ ; that is, the line is parallel to the axis of Z. 

Examples. — Find the equations of the line passing through 

(1, 2, 1) and (3, — 2, 1) ; through (1, 2, 1) and (1, 3, 1). 

385. When L, M and N have finite values, the equations of the 
line may be reduced to the forms y = mx -f- b and z = nx -}- c, in 



EQUATIONS OF THE STRAIGHT LINE. 281 

M N 

which m = — and n = —. In these equations, b and c determine 
L Li 

the point in which the line cuts the plane of YZ, for the point 

(0, 6, c) satisfies the equations. 

In finding the intersection of a straight line with a given plane, 
or the point common to the line and the plane, we have to combine 
the two equations of the line and the equation of the plane, in 
order to determine the values of the three co-ordinates. This is, in 
fact, the same thing as combining the equations of three planes 
to find their common point, or the values of x, y and z : which 
satisfy them simultaneously. It is most convenient to express the 
values of y and z in terms of #, for the equations of the line, and 
then to substitute them in that of the plane. Thus, given the line 
y = 3x -j- 1, z = — x -\- 2 and the plane 2x -\- y -j- 3z — 5 = 0, 
we find by substitution 2x -j- 2 = or x = — 1. The corre- 
sponding values of y and z, determined by the equations of the line, 
are y = — 2, z= 3. Therefore the required point is ( — 1, — 2, 3), 
as may be verified in the equation of the plane. 

When two straight lines are given by their equations, we have in 
reality four equations of planes. Therefore, to ascertain whether 
the lines intersect, find the values of x, y and z. which satisfy three 
of the equations ; if they satisfy the fourth equation also, the lines 
intersect. 

Examples. — Determine the intersection of the line passing 
through (5, — 3, 2) and (1, 1, — 6) with the plane x — 3y -j- z -\- 
10 = 0; of the same line with the plane of XY. 

Does the line joining (1, 3, 1) and the origin cut that joining 
(2, 1, — 1) and ( — 1, 2, 4) ? Does either of these lines cut the 
line y = 2, z = 5 -\- x? 

386. Substituting y = mx -\- b, z = nx -j- c in the general 
equation of the plane, Ax -f- By -j- Cz -j- D = 0, we have 

(A -j- Bm -f Cn) x -f Bb -f Cc -f D = 0. 

If A -L. B??? -f- Cn = 0, the value of x derived from this equation 
generally takes the infinite form, which indicates that the line and 
plane have no common point; that is, that they are parallel. (But 
if at the same time Bb -\- Cc -(- D = 0, x takes the indeterminate 
form, which indicates that the line is situated in the plane.) Sub- 
24* 



282 SOLID GEOMETRY. 

M N 

stituting for m and n their general values, — and — , we have, for 

L L 

the condition that a line and plane shall be parallel, 
AL + BM + CN = 0. 

Thus, the line == J = is parallel to the plane 

O o — L 

x+y-\-4z = 0. 

387. If in the equations of Art. 383 the axes are rectangular, 
the denominators x" — a?', y" — y' and z" — z' are the projections 
of P"P' upon the axes. Therefore, we may substitute for these 
quantities P"P' cos a, P"P' cos /? and P"P' cos y. If we then mul- 
tiply each member of the equation by P"P', the result is 

x — x r y — ?/ z — rf 



cos a cos ft cos y 

These are the equations of the line passing through P', and having 
the direction angles a, ft and y. Each of the three members is 
a value of PP', the distance between any point of the line and the 
fixed point P.' 

To reduce to this form the equations of a line given in the gene- 
ral form of Art. 384, we must divide the values of L, M and N by 
"|/_L 2 -\- M 2 -f~ N 2 - We therefore find the direction cosines of a 
line in a manner similar to that in which we found the direction 
cosines of the perpendicular to a plane in Art. 376. 

388. In the general rectangular equations of the straight line 
and plane, L, M and N are proportional to the direction cosines 
of the line, and A, B and C to those of the perpendicular or axis 
of the plane. The condition found in Art. 386 may therefore be 
regarded as expressing that the line is perpendicular to the axis of 
the plane. See Art. 378. The condition for perpendicular lines, 
found in a similar manner, is 

LL'-fMM' + NN' = 0. 

To be perpendicular to a given line, or to be parallel to a given plane, 
is equivalent to a single condition imposed upon the ratios of L, M 
and N j therefore two such conditions may be fulfilled. But, to be 



EQUATIONS OF THE STRAIGHT LIXE. 2S3 

parallel to a given line, or to be perpendicular to a given plane, 
determines the ratios of L, M and N : because these quantities must 
then be proportional to the given values of L, M and X, or to those 
of A, B and C. 

389. When the axes are rectangular, the distance of a point 
from a given point, plane or line may readily be found. Since the 
square of a line is equal to the sum of the squares of its projections 
upon the axes, the formula for the distance between two points is 



FT' = V{z" - xj + (y" -y'f + (f - z'f. 

hetp' denote the distance or perpendicular from P' to the plane 
x cos a -j- y cos ,3 -J- z cos y — p = 0. Now the projection of OP' 
upon the perpendicular to the plane, found as in Art. 376, is 
re' cos a $f- y cos /5 -j- z 1 cos y. The difference between this projec- 
tion andp is equal to p' . Hence 

p' — x' cos a -|- y' cos ,3 -j- 2/ cos y — p. 

Therefore, to find the perpendicular distance from a point to a 
given plane, reduce its equation to the above form (dividing by 
the square root of the sum of the squares of the coefficients), and 
then substitute the co-ordinates of the point in the first member. 
Letp" denote the distance or perpendicular from P" to the line 

— ■ = '■ . Let a plane perpendicular to the 

cos a cos /3 cos y 

given line be passed through P": it will contain the perpendicular 
p". Let j?' denote the portion of the given line intercepted be- 
tween this plane and the known point of the line, P' ; that is, the 
perpendicular from P' to the plane. Then p" and p' will be the 
sides of a right triangle whose hypothenuse is the distance of the 
points V" and P' ; therefore p' n = P"P' 2 — p'\ The equation of 
the plane is, by the formula of Art. 379, (x. — x") cos a -f- 

(y — y") cos fi + \ z — z ") cos r = °- The Yalues of p' and p ' /pr 

may be found by the formulae given above. Thus, given the point 
(4, 2, 1) and the line = =2 — 2; the direction co- 

_ L 

sines of the line are -J, -|, -J, and the equation of the plane is 
%(x — 4) -f I (y — 2) -f i — 1) = 0. The perpendicular 



284 SOLID GEOMETRY. 

from F, (3,-1, 2), is f, and the distance FT' is j/11 j hence 
P " = VuT=^f= i t/50 = | x/-2. 

Surfaces of the Second Degree. 

390. The surface generated by the rotation of any conic section 
about one of its axes is called a surface of revolution. The equa- 
tion of a surface of revolution may be found, when that of the 
generating curve is known, in the following manner : 

Let the surface be referred to rectangular axes, the line about 
which the rotation takes place being made the axis of Z. The 
section of the surface by a plane passing through the axis of Z will 
then be the generating curve. Let r denote the distance of a point 
of this curve from the axis of Z ; then r and z are the rectangular 
co-ordinates of P, and the equation of the curve is a relation be- 
tween r and z. Since the axis of Z is an axis of the curve, the 
equation will not contain the first power of r. Now by Art. 354, 
r 2 — x' 2 -f- y 2 ; and since we have a relation between r 2 and z, which 
is true of every point of the surface, if we substitute x 2 -\- y 2 for r 2 , 
the result will be the equation of the surface. The equations of the 
surfaces thus described are therefore of the second degree. 

For example, r 2 = 2pz is the equation of a parabola whose axis 
coincides with the axis of Z (z taking the place of x, and r that of 
?/, in the ordinary equation y 2 = 2px). Therefore x 2 -j- y 2 = 2pz 
is the equation of the surface generated, which is called the para- 
boloid of revolution. The trace of this surface upon either the 
plane of XZ or that of YZ is a parabola having its vertex at the 
origin. The section by the plane z = c is a circle whose radius is 
~V%pc, which is real when c is positive, and imaginary when c is 
negative. 

391. The surface of a right cone is described by a straight line 
rotating about a fixed line, which it cuts at a constant angle. 
Taking the origin at the intersection of the generating line with 
the fixed line or axis of Z, the equation of the line is r = mz, in 
which m is the tangent of the constant angle. Squaring we have 
r 2 = m 2 z 2 ; therefore the equation of the cone is 

x 2 -j- y 2 = m 2 z 2 . 
The trace upon the plane of YZ is y 2 = 7n 2 z 2 or y = ± mz. It is 



SURFACES OF THE SECOND DEGREE. 285 

therefore a pair of straight lines making equal angles with the axis 
of Z. These lines, in fact, constitute a conic of which an axis coin- 
cides with the axis of Z. The section by the plane z = c is a cir- 
cle whose radius is mc, which is always real ; but when c = 0, the 
circle reduces to a single point. 

If the generating line be parallel to the axis of Z, the surface 
generated will be that of the right cylinder. The equation of the 
generating line will then be r = b; and that of the cylinder is 

x 2 -f f = b 2 . 

The equation of the surface of the sphere, described by the rota- 
tion of the circle r 2 -j- z 2 = B, 2 , is 

x 2 +f + z 2 = W, 

in which R denotes the radius of the sphere. 

392. When the rectangular equation of a surface contains only 
the squares of the variables and an absolute term, it can be put in 
the form 

± A 2 "*" ± B 2 "*" ± C 2 

The trace of the surface upon either of the co-ordinate planes will 
then be an ellipse or hyperbola referred to its centre and axes. If 
the denominators are all positive, the traces are all ellipses. The 
surface, in this case, is called an ellipsoid. This surface encloses a 
space, and cuts each of the co-ordinates axes. The intercepts A, B 
and C are called the semi-axes of the ellipsoid. If two of the semi- 
axes are equal the ellipsoid becomes a spheroid : it is called & prolate 
or an oblate spheroid according as the third semi-axis is greater or 
less than either of the others. If all three semi-axes are equal, 
the equation reduces to that of the sphere. 

If two of the denominators are positive and the other negative, 
two of the traces upon the co-ordinate planes are hyperbolas, and the 
other is an ellipse. The surface, in this case, cuts two of the axes, 
but not the third. It is a continuous but not a closed surface, and 
is called an hyperboloid of one nappe. If the two positive denomi- 
nators are equal, the surface is that generated by the rotation of an 
hyperbola about its conjugate axis. 



286 SOLID GEOMETRY. 

If one of the denominators is positive and the other two nega- 
tive, two of the traces are hyperbolas, and the third is an imaginary 
ellipse ; that is, the surface does not intersect the third plane. The 
surface then consists of two distinct parts, one on each side of this 
plane ; it is therefore called an liyperboloiol of two nappes. It in- 
tersects only one of the axes. If the two negative denominators 
are equal, the surface is that generated by the rotation of an hyper- 
bola about its transverse axis. 

If all three of the denominators are negative, the traces are all 
imaginary and the surface disappears. 

393. The surface represented by an equation of the above form 
is symmetrical with respect to each of the co-ordinate planes; for, if 
we assume values of two of the variables, the equation gives equal 
positive and negative values of the third. The surface is therefore 
said to be referred to its centre and axes. 

The equation of a surface of the second degree may generally be 
reduced to this form, by transformation of co-ordinates. In the first 
place, the axes may be made rectangular ; then the three terms con- 
taining the products of the variables may be made to disappear by 
change in the direction of the axes, for, by Art. 367, the formulae 
for passing from one rectangular system to another contain three 
arbitrary constants. This simplification of the equation is found to 
be always possible. Then, if the result contains the square of each of 
the variables, the terms of the first degree may be made to disappear 
by change of origin, using the formulae of Art. 365. If now the 
reduced equation contains an absolute term, it represents one of 
the central surfaces named in the last Article. But if it takes the 
form 

Ax 2 + By 2 + (V = 0, 

the surface may be regarded as the vanishing case of an ellipsoid 
or of an hyperboloid, according as the coefficients are all of the same 
sign, or two only of the same sign. In the first case, the equation 
is satisfied only by the origin. In the second case, the traces on 
two of the co-ordinate planes are pairs of straight lines passing 
through the centre, and the sections parallel to the third plane are 
similar ellipses. The surface in this case is that of a cone. When 
the coefficients of the same sign are equal, it is a right cone or cone 
of revolution. 



SURFACES OF THE SECOND DEGREE. 287 

394. When the equation, after being freed from the products of 
the variables, contains the squares of only two of the variables, the 
term containing the first power of the other variable cannot be 
made to disappear. Suppose, for example, that the equation does 
not contain 2 2 , then the term containing z cannot be made to van- 
ish. If this term exists in the equation, we may, however, so deter- 
mine the constant of transformation, z\ as to make the absolute 
term vanish. The equation will then take the form 

Ax 2 -f Bif -f Kz = 0. 

The traces of this surface on the planes of XZ and YZ are para- 
bolas whose axes coincide with the axis of Z. If A and B are of 
the same sign, these parabolas are turned in the same direction, and 
the sections by planes parallel to the plane of XY, on one side of 
it, are real ellipses. The surface, in this case, is called the elliptic 
paraboloid. When A = B, the sections are circles, and we have 
the paraboloid of revolution. 

When A and B are of opposite signs, the axes of the parabolas 
have opposite directions. The trace on the plane of XY is a pair 
of straight lines, and the sections parallel to this plane are hyper- 
bolas. The surface is therefore called the hyperbolic paraboloid. 

The paraboloids are called non-central surfaces. They are sym- 
metrical with respect to two of the co-ordinate planes. The inter- 
section of these planes is the axis of the surface : in the equation, 
this line is the axis of Z. 

If the term containing z does not occur in the equation which 
we have supposed not to contain z' 2 , the equation has the form 

Ax 2 -f By 2 -f L = 0. 

The traces on the planes of XZ and YZ are now pairs of parallel 
lines. The trace upon the plane of XY is an ellipse or an hyperbola, 
according as A and B have the same or opposite signs ; and the 
sections by parallel planes are all equal to the trace. The surface, 
in this case, is that of an elliptic cylinder or of an hyperbolic cylin* 
der. The axis of Z is the axis of the cylinder, and any point on 
that line may be taken as the centre of the surface. These cylin- 
ders are therefore said to have a line of centres. 



288 

SOLID GEOMETRY. 



■^iE££S5£2r , to a straight line - and *• 

395. When the rectaZT ♦ mg '" the axis of & 

one of the n^^^-Jf- ^ H « *«* 
made to disappear. We mav in 1 WmWe onI J ca " be 

Az 2 + % + Kz = o. 

The traces oa the planes of XY and T7 

on the pl ane of Yz is , J^ton** are P ar abolas. The trace 
and the sections by paralle £! ^T 8 thr ° U S h the °"P», 

to the trace. B«Sfe^^*-pi 
This snrface is symmetrical wi h f " *" ***** ***** 
nate planes. J ^ ^P 60 ' to °«e only of the co-ordi- 

thi tS;r;tL* e fat **"<0- .-well as 
form g SqUareS ' are ^ntmg, the equation has the 

Aa 2 -f 1 — o. 
This equation represents a pair of real or in,, • 
aecording as A and L have the^JT*'™? 11 ™*™*'** 
Parallel planes constitute a s„ f ace «f wnTT BigM - TheSe 
taken anywhere in the plane of YZ If I n T" ™* be 
coincident. irL = 0, the planes are 

th- cylinders and pairs "pW^n 2 ^T^^ the 
of the surface of the second degrel ^ ^ 0n, ^ a ™ties 



